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Full text of "A Text Book of Heat"



A TEXT BOOK OF HEAT 

FOR JUNIOR STUDENTS 

(INCLUDING KINETIC THEORY OF GASES, 
THERMODYNAMICS AND RADIATION) 



BY 

M. N. SAHA, ftSc, F.R.S., 
Late Palit Professor of Physics, Calcutta Universtty 

AND 

B. N, SRIVASTAVA, D.Sc^ F.N.I., 

Professor of General Physics, Indian Association for the 
Cultivation of Science, Calcutta 



(TWELFTH EDITION) 



SCIENCE BOOK AGENCY 

I*. I'i'-ili, Lake Terrace, CaIcutta-29 



First Edition 




Second Impression 




Sf.co.nd Edition 


1939 


Third Edition 


1943 


Fourth Edition 


1945 


Fifth Edition 


1949 


Sixth Edition 


1951 


Seventh Edition 


1953 


Eighth Edition 


1954 


ON 




1'iuntr Edition 




ELEVENTH !-'n 




Twelfth 





ill Rights resen 

Publishku by Sm, Roma Saita, ] : i-hern Avenue, 

Calcutta : 

Printed by Modern India Press 
7, Raja Sudodh Mullick Square, Cai-cutta 13 

Price Rupees Nine only 






PREFACE TO THE FIRST EDITION 



i Text. Book of Heat for Junior Students has been written with 

iv to supplying the needs of the students of the pass course 

ir the Bachelor's degree. It has grown out of the lectures 

which i he senior author has been giving to die B.Sc, pass class of 

Ulahabad University for several years. The plan of 
closely follows that of the larger Text Book which is intended for 
B.Sc. honours and M.Sc. students. Separate chapters have been 
i i voted to Kinetic Theory, Liquefaction of Gases and Heat. Engines, 

principles of Thermodynamics and their applications have been 
Created at considerable length. Throughout the book the methods of 
calculus have been freely employed. The supplementary chapter on 

orology has been kindly written by Dr. A. K. Das of the Indian 
Meteorological Sendee and Mr. B. N\ Srivasuava. Meteorology is a 
growing science and is extremely useful to die public at large. It is 
not at present included in the curriculum of any Indian University 
il Agra where it forms a special course for the M«Sc. degree) , 
hut this seems to be a cardinal omission. It is hoped that in future 
it will form a regular subject of study by degree students, 

\& this is the first edition of the book, it is feared that there 
may he several omissions and inaccuracies. The authors will be 
grateful if these arc brought to their notice. 

Allahabad : M. N, S. 

maty, 1933. B* N. S. 

PREFACE TO THE ELEVENTH EDITION 

Since the last edition many Indian Universities have Introduced 
the new three-year Degree Course while some others are still continu- 
ing the old two-year course. The book has therefore been thoroughly 
revised to cover the syllabus of the new three-year course of most- 
Indian universities. As this required only addition of some matter 
previously found in the Intermediate Syllabus and as the old subject 
matter of the book has been almost wholly retained, it is confidently 
hoped that the book will prove equally useful both for the new three- 
course and the old two-veai course. 



Calcutta : 

Juk t 1962. 



B. N. S. 



PREFACE TO THE TWELFTH EDITION 

Several suggestions for the improvement of the book, kindly 
i hi by teachers using the book, have been incorporated in this 
edition. The .standard questions, arranged chapterwise and given at 
the end o! the book have been brought uptndate. 

I. una : 
luajy, J 967. B, N, & 



ACKNOWLEDGMENTS 



CONTENTS 






We have much pleasure in expressing our indebtedness to the 
following authors, publishers and societies for allowing us to reproduce 
diagrams which appeared in the works mentioned below : — 

Muller-Pouillets Lehrbuch der Physik, 

Heading, Temperat itrmessung, 

Nernst, Grundlagen des neuen Warmesatzes. 

Jellinek, Lehrbuch der physiftalischen Chemie. 

Ewing, The Steam Engine and Other Heat Engines. 

Watson, Practical Physics. 

Burgess and Le Chatclicr, The Measurement of High Temperatures. 

Proceedings of the Physical Society of London, 

Physical Review. 

Journal of the Optical Society of America, Vol. 10. 

Proceedings of the Royal Society, London, Chap, II, Figs, 4, 5 ; 
Chap. VII, Fisr. 10. 

Philosophical lions of the Royal Society, Chap. II, Fi 

Char, vn, Fig, io. 

Zeiti , V, Fig, 

: ; , Chap, i I Fig. 16. 
Atmai >' ; Chap, VII, Fig. 3. 

[buck d< Chap, IV. Fig. 7; Chap. VI, Fig. 10; 

Chap. XL Fig. 
Handbuch der Experi physik. Chap. V, Fig. 11; Chap. VI, 

Pigi tap. Vni, Fig, ' hap. XT, Yi^. 11, 18, 19, 20. 

Ezer I (hods of Measuring Temperature. Chap. I, Figs. 

5, 6, 7, 9; Chap. XT, Figs. M. 15, 21, 22, 23 from diagrams on 
52, 52, 34, 71. 84, 84, 90, 90, 115 ol the work respectively. 

Andrade, Engines, Chap. VI, Fig. 5 ; Chap. TX, Figs. 1, 3, 4, 
7, 8, IS, 16, 17 respectively from pages 239, Gl, 68, 75, 102, 86, 
92, 107, 195, 211, 213 of the work by the kind permission of 
Messrs. G. Bell & Sons, Ltd, (London) . 

Partington and Shilling, Specific //eats of Gases. Chap. II, Figs. 13, 
15, 17. respective:];, from pages 127, 76., B4 of the work. 

OH, Theory of Heat. Chap. I, Fig. 3; Chap. II, Figs. 3, 12; 
Chap. XI, Fig. 16, Reproduced by the kind permission ol 

Messrs, Macmillan & Co. 



Chapter 



Pace 



1. Thermometry 

Temperature. Mercury thermometer. Special types of liquid 
thermometers. Gas thermometers, Callendar compensated air- 
thermometer. Standard gas thermometers. Perfect gas scale, 

.Jardization of secondary thermometers. Fixed temperature 
Platinum thei era. Measurement of resistance. 

mo-couples. Low temperature thermometry. International 
temperature scale. Illustration of the i i t Thermometry, 

II. Calorimetry 

Quantity of heat Methods in calorimetry. Method of mixtures. 

Radiation correction. Specific heat of solids. Specific heat o£ 

liquids Method of cooling. Method of melting ice. BunserVs 

ice calorimeter. Joly's steam calorimeter. The differential 

steam calorimeter. Methods baser! on the rise of temperature. 

oil of steady-flow electric calorimeter. Specific heat of 

:ter, Results of early experi- 

■ arit. t. ion oi with temperature Two specific 

. Experiments of Gay-Lussac and Joule. Adiabatic 

transformations. I- ital methods. Method of mixtures, 

method. Measurement of C. Explosion meti 

Wiakitie i^xpansirn method). Experiments 
of Clement and Desormes. Experiments of Partington. "Ruchardt's 
experiment. Veloei und method. Kundt's tube. Expe- 

riments of Partington and Shilling. Specific heat of a 
incur, Results, Special calorime 1 

III. Kinetic Theory of Matter 

The nature of heat Joule's experiments, I:. ■• "i in 

mics, Metl ermrning /, !••!".. ai i nnciits. 

i: ;>:: laboratory method of finding- /, Electrical methods. 
Heat as n I m decides. Growth of the kinetic tl 

Evidence of molecular agitation. Brownian movement Pn 
ted by a perfect gas. J : f gas laws. Introd 

of temperature. Distribution of ■. I .'.veil's law. 

cities. Law of equlpartition of kinetic energy. 
■iilrir and atomic Mean free path phenomena. 

:: of the mean tree path. Transport phenomena.* 
Viscosity. Conduction, Value ol coi 

[V. Equations of State for Gases 

iation from the perfect gas equation. Andrews' experiments, 
ai der WaaU* ei of state. Methods uf finding the 

values of V and 'b'. Discussion of van der Waals' equation. 
Experimental study of the equation of state. Experimental 
determination of critical constants. Matter near the critical 
point. 

V. Change of State 

Fusion. I -a tent heat. Sublimation Change of properties on 

ting. Determination of the melting point. Determination 

of the latent heat of fusion. Experimental relationships. 



104 






VI 



Chai 



l*Afi» 




lances at liquid 
technique. Uses of 
The air conditioning' 



:. melting point. Fusion of 
cooling, boiling and superheating. Saturated ami 

unsati ■ lira Vaj <ur pressors of water. Statical 

jethods. Dynamical method. Discussion of results, Vapour 
pressure over curved surfaces. Latent heat of vaporization. 
Condensation methods. Evaporation methods. Variation of 
latent heat with temperature, Trouton's rule. Determination 
of vapour density. Accurate determination of the density of 
saturated vapour, 

VI. Production of Low Temperatures .. ,-125 

Principles used in refrigeration. Adding a saft to ice. Boiling 
a liquid under reduced pressure. Vapour compression inachine. 
Refrigerants. Electric refrigerator. Ammonia absorption 
machine. Adiabatic expansion of compressed gas ooling 

due to Peltier effect. Cooling due to desorption. Liquefaction of 
gases. Liquefaction by application of pressure and low tempe- 
rature. The principle of cascade-,. Production of low tempera- 
tare by utilizing the Joule-Thomson effect Elementary heory 
of the porous iperimeui. The porous plug experiment. 

Principle of regenerative cooling. Lilidc's machine for liquefying 
air. Hampson's air h'queficr. Claude's air liqueficr, Liquefac- 
tion of helium. Solidification of helium. Cooling produced by 
adfabatic demagnetisation. Properties of 
helium temperatures. Low temperature 
liquid air. Principles of air eondi 
hine. 

VII, Thermal Expansion .. 157 

ansion of solids. Lii rrlfcr measurements 

of liner Camparatoi method, 

ihod. Fizeau's interference method, 
fringe ter. Surface and volume expansion, 

Expansion of [ass, invar. Expansion of ■ 

bodies. Expansion of liquids. The difatometer method. The 
weighty thermometer mctlmd. Matthiesscn's method, Absolute 
expansion of Hqiuds. Hydrostatic balance method, Expansion 
of mercury, water. Practical app 

pensation of clocks and watches. Thermostats. Expansion of 
Rases. t Experimental determination of the volume coefficient of 
expansion. Experimental determination of the pressure co- 
efficient of expansion. 

VIII, Conduction of Heat . . . . . . 178- 

Methods of heat propagation. Conductivity of different kinds 
of maiter._ Definition of conductivity. Conductivity of mewls. 
Conductivity from calorjmetric measurement. Rectilinear flow 
"f heat. Mathematical investigation. Ehgen-Hausz^s experi- 
ment. Experiments of Despretej Wiedemann and Franz. Forbes' 
rcieUi I'om's metliod. Conductivity of the Earth's crust. 

Conduction through composite walls. Relation between the ther- 
mal and electrical conductivities of metals. Heat conduction in 
• dimensions. Conductivity of poorly conducting solids. 
Spherical shell method. Cylindrical shell method. Lees' disc 
method. Conductivity of liquids. Column method. Film method. 
Conductivity of gases. Hotwire method, Film method. Results. 
Freezing of a pond Convection of heat. Natural convection- 



VI 1 



IPTER 



IK. Heat Engines 

Three different classes of jngines. Early history of the steam 
engine, bfewcomen's atmospheric engine. James Watt. fJse 
oi a separate condenser, The double-acting engine. Utilisation 
of the expansive power of steam. The governor and the throttle 
valve. The crank and the flywheel. Modern steam engines. 
Efficiency oi engines and indicator diagrams. The Carnot 
engine. Reversible and irreversible processes. Reversibility of 
the Carnot cycle Carnot's theorem. Rankhie cycle. Total heat 
of _ steam. Internal _ combustion i engines. The Otto cycle, 
1 scl cycle. Semi-Diesel engines, The 'National' gas 
engine. Diesel four-stroke engines. The steam turbine, The 
theory of steam jets. The De Laval turbine. Ratcau and Zolly 
turbines, Reaction turbines (Parsons}, Alternative types of 
engines. Thermodynamics of refrigeratinn. Efficiency of a 
vapour compression machine. 



XI. 



Page 
204 



X. Thermodynamics . . 



.. 2c57 



Scope oi thermodynamics. The thermal state of a body. 
Mathematical notes Some physical applications. Different forms 
of energy. Transmutation of energy. Conservation of energy. 
Dissipation of energy. The first law of tiicmtodynamics. 
Applications of the first law. _ Specific heat of a body.' Work 
done hi certain processes. Discontinuous changes in energy — 
latent heat Mess's law of constant heat-summation. Second 
>f thermodynamics and entropy. Scope of the second law. 
Preliminary statement of the second law. Absolute scale of 
temperature. Definition of entropy. Entropy of a syg 
Entropy remains constant in reversible processes. Entropy in- 
creases in irreversible processes. Entropy of a perfect ' gas. 
General statement of second law of thermodynamics. Supposed 
violation of the second law. Entropy and unavailable energy. 
Physical concept of entropy. Entropy— temperature diagrams' 
Entropy of steam. Applications of the two laws of thermo- 
dynamics, The thermodynamics! relationships (Maxwell). First 
on. Application to a liquid film. Second relation. Other 
itions. variation of intrinsic energy with volume, foute- 
Thontson effect. Correction of gas thermometer. Examples. 
Clapeyron's deduction of the Clausius-Clapevron relation, Speci- 
fic heat of saturated vapour. The triple point. 

Radiation 

Some simple instruments for measuring radiation. Properties 
and nature of radiant energy. Identity of radiant energy and 
hght— continuity of spectrum. Fundamental radiation processes. 
Theory of exchanges (Prevost), Laws of cooling. Emissive 
power of different substances. Reflecting power. Absorption, 
itions existing between the different radiation quantities 
Fundamental definitions, Kirchhoff's law. Applications of 
Kifchhoff s law, Application to astrophysics. Temperature 
radiation. Exchange of energy between radiation and matter in 
a hollow enclosure. Deduction of Kirchhoff's law. The black 
body. Radiometers! Sensitiveness of the thermopile. Crnokes' 
radiometer. Bolometer. Kadiomicrometer. Pressure of radia- 
tion. Total radiation from a black body— the Stcfan-Bollzmann 
law. Experimental verification of Stefan's law. Laws of dis- 
tribution of energy in blackbody spectrum. Experimental study 
ot the blackbody spectrum. Pyrometry. Gas pyrometers. Resis"- 



278 






VIII 



Chapter 






Page 



XII. Thermodynamics o> the ATSfosPHERE 



Examples (I to XII) 
Answers to Numerical Examples 
Table of Physical Constants 
Subject Index 



316 



327 
331 

349 

351 
352 






CHAPTER I 
THERMOMETRY 

1. Temperature.— The sensation of heat or cold is a matter 

t ail} 1 experience. By the mere sense of touch we can say whether 
a substance is hotter or colder than ourselves. The hot body is said 
to possess a higher temperature than the cold one. 

But the sense of touch is merely qualitative, while scientific 

ecision requires that every physical quantity should be measurable 

numerical terms. Further, the measurements must be accurate 

easily reproducible. This requires that the problem should be 

handled objectively and tile sense of touch should be discarded in 

favour of something which satisfies the above criteria. Let us sec how 

can be done. 

When two bodies are brought in contact, it is found that, in 

general., there is a change in their properties such as volume, pressure 

etc. due to exchange of heat. Finally an equilibrium state is attained 

ter which there is no further change. The two bodies are then 

said to be in thermal equilibrium with each other. In this state of 

M, I equilibrium the two bodies are said to have the same tempe- 

which ensures their being in thermal equilibrium, Also it is 

[ound that if a body A is in thermal equilibrium with two bodies B 

and C, then li and C will be in thermal equilibrium with each other 

therefore be at the same temperature. These are the two 

fundamental laws of thermal equilibrium and it is on account of these 

s that we arc able to measure the temperature of bodies B and C 

bringing them successively in thermal equilibrium with the thermo- 

neter A. The temperature of a system is a property which determirii 

whether or not a system is in thermal equilibrium with other systems. 

Heat causes many changes in the physical properties of matter 

some of which are well known, e.g., expansion, change in electrical 

.laiire. production of electromotive force at the junction of two 

iniilai' metals. All these effecLs have been utilised for the 

isurement of temperature. The earliest and commonest therm o- 

titilise the property of expansion. Mercury-in-glass is univets- 

em ployed as a thermometer for ordinary purposes, but though it 

iiiplc. convenient to use and direct-reading, it is not sufficiently 

for high-class scientific work. 

Mercury Thermometer.** — Everybody is familiar with the 
• • i ■. 1 1 1 - . 1 1 ' centigrade thermometer. It consists of a glass bulb con- 
mercury to which a graduated capillary stem is attached. 
The freezing point of water is marked 0°C and the boiling point 100°C 
the interval divided into 100 equal parts. This scale was first 
i In >1 by Celsiusf and is called the Celsius or centigrade scale 

I r details of construction see Preston, Theory of FIcat, Chapter 2. 
\ A i O -us (1701-1744) was borti at Upsala where he studied matbe- 

tlci ii I astronomy. In 1750 he became Professor of Astronomy and ten years 
hr built the observatory at Upsala and became its director. He invented 
'"ale. 




THERMOMETRY 






and is now adopted for all scientific work. Other stales in ordinary 
Use today are thos€ introduced by Fahrenheit and Reaumur. But, 
Fahrenheit* was the first to choose mercury as die thermoinetric 
substance cm account of its many advantages. It does not v, 
can be easily obtained pure, remains liquid over a fairly wide range, 
has a low specific heat and high conductivity ; it is opaque am 
expansion is approximately uniform and regular. But we must not 
orget its several drawbacks. The specific gravity and surface tension 
of mercury are large, and the angle of contact with glass when 
mercury is rising is different from that when it is falling-. On account 
of these defects alcohol is sometimes used in place of mercury, and 
since it has a larger expansion it is more sensitive but. is likely 
to distil over to colder pans oi the tube. 

The range of an ordinary mercury thermometer is limited by 
the fact that mercu: tes at -38.8°C and boils at 856 °C but 

the upper limit can be raised to about 500 "C by filling the top of 
the tube with nitrogen under pressure. The thermometric glass must 
be of special quality ; it should be stable and should rapic urn 

to its normal state after exposure Co high temperatures. The gla 
generally employed are v .■• and Jena It'. 1 " for ' her- 

mometer; orosilicate glass 59 111 for high-temperature work. 

Mercury thermomel generally o.d for rough work. 

If tin at all fork various corrections must 

be applied to get tJi: mpcrature. Tlie important ones am 

ihe : — 

This is very in po i ini 
sine high tempers mal 

ion due to change in the fundamental i om 

to 100-j-S (say) I i lue. 

ition Con his is due to want of uniformity 

in the bore of the capillary n 

(4) Correction for lag of the thermometer. This increases with 
!ze uf the bulb, 

(5) Correction due to changes in the size of the bulb caused by 
variable internal and external pressure, 

(6) Correction for the effects of capillarity. 

Exposed stem correction. Part of the stem and hence the 
contained mercury does not acquire the temperature of the bath. 

For details concerning the application of these corrections 
Appendix I. 

3. Special Types of Liquid Thermometers. — The ordinary mer- 
cury thermometer is not suitable for certain purposes ; for this 
reason special types of thermometers have been devised, are 

"Daniel Gabriel Fahrenheit l b 16-1736) was born in Danzig of a rid fa 
tmstenktn. He made improvements in the t- 
ised his '.. ric scale. 






SPECIAL TYPES OF LIQUID THERMOMETERS 



of the ordinary thermometer designed to serve the 

\ lew, 

cological purposes thermometers arc required to 
! cate the maximum and minimum temperatures to which they 
osed during a certain period. Six devised a combi 
mum and minimum thermometer which is indicated in Fig. L 
II if and part of die tube is filled with alcohol up to the level 

in with mercui") up to C above which again there is alcohol 
■ glass indexes I, I, have each an iron wire attached 
(shown separately), and placed above B 
Hid CI in each tube. When the tempera- 
es i lie alcohol in A expands and 
landing mercury, on account of its 
tension j pushes "upwards the index 
ie C to its maximum limit. With a 
I j II in temperature this index is undis- 
turbed due to the viscosity of alcohol being 
II while the index above B is pulled 
1 by the contracting alcohol, but is 
I. n 'hind when the temperature rises, 
iron wire attached to die glass in: I .: 
prevents it from falling under its own 
In. and enables its position to be 
I From outside. 

n ordinary 

rem -y thermometers oE 

I the maximum type. The stem 

the bulb has a constriction 

■I which the mercury passes when 

iis temperature rises. On cooling, the 

iny is unable to force its way bark, 

I lie range of temperatures is usually. 95 °F 

i I I:) r and the bulb is very thin and 

capillary bore very fine. The mercury 

thread is rendered easily visible by con- 

. 1 1 1 1 « I i 1 1 g 1 ens-f ron t thermomc t ers. 

For accurate work, such as the determination of the boiling and 
[ting points of organic substances, several short-range thermo- 
• era employed between the range and 490°C. They are 

U thermometers. Benzol and toluol thermometer 
1 1 1 1 > 1 1 • ■ ■ i the many that are in use. 

The Hechmann thermometer, indi I in fig. 2, is used to 
ni> nail changes of temperature with a high degree of accuracy. 

is here marked from to 5 representing approximai 

centigrade degrees and even? degree is di into 100 equal parts. 

.: reservoir at the top of the instrument, shown separately 

H e the range to be varied. To set the thermometer lo 

ii' desired range ihe bulb is heated to c n rcury into the reser- 



D A 




• • ,■}'] 


:r. 






Lib 


:,.. 






*o 


- ' 






IB 


10 ' • ' V 


1 


10 

'I 


1 | , 


• 


OH 


- 16 




i'B 


■'•• 


' ' i"" 


!-J 


1 ■■'-.■ 


ifi. 


(\ 


E6 


"j 


1 r 


go 


"^^^ 



Fig. L — Six's maximum and 
imum thermometer. 



THERMOMETRY 



[CHAIV 



voir and the instrument gently tapped when the mercury column 
breaks near the reservoir and some mercury i 
transferred into it. Next the Beckmarm tin 
meter is immersed along with an auxiliary 
thermometer in a bath whose tempera tun 
varied till the mercury stands at division "l 
the former. The temperature corresponding to 
the aero of the Beckmarm thermometer is thus 
t -I v-i -i v e 1. 1 on 1 1 1 e a uxiliary th ermom eter, and b) 
varying the amount of mercury in the bulb this 
is adjusted to be near the desired range. The 
value of each scale division varies with the 
quantity of mercury in the bulb and a correc- 
tion curve Lor different settings of the zero is 
supplied with the instrument from which the 
correction at any point of the scale ma) bi 
obtained. 

4. Gas Thermometers. — The fundamental 

vantage of liquid them; sters is that two 

thermometers containing differenl Liquids 
as mercury and alcohol, and graduated as on 
page 1 will agree probably only at.0°C and 100°C 
ana at no other temperature. This is due to the 
the expansion of the two liquid; is not 
regular and similar, Thus the mercury thermo- 

page I would give an arbitrary* 6 
of temperature. Moreover, the corrections to 
iplied to it ■:}■■ 2) are uncertain and known 
ly. Hence, tor accurate work 
iry thermometers arc calibrated (see Sec. 8) 
lual comparison with a resistance tin 
meter throughout the entire range. Even then 
the Riercur) thermometer is rarely used for 
accurate work, and [or all standard work pas 
mometcrs are employed. 

Gases •• i/« I servants 

i mometric substance. Their expansion is 
large so that gas thermometers will be more 
sensitive and the expansion of the containing 

I will necessitate only a very small correc- 
tion, rheir expansion is- also regular, i.e., the 

nsion of a volume of gas at 0°C is the same 



W 



2< — Becfcmauu 
thermometer. 



•The v mercury at any temperature i lying 

between and IQO'-'C as measured on the perfect gas scale is given by the r 
Pi - Bd(1 + 1-B182 x I0-'.' + 078 X 10-" t*)< 

- the relation ts not linear and the readings of the mercury thermometer 
iratod on p. I will not agree, even after applying the corrections mentioned 
1 1 i A a perfect giv .. cter even in the range 0°— 100°G 




1. 1 GAS rHURMOMETERS 5 

cry one degree rise of temperature. They can be obtained pure 
and remain gaseous over wide ranges of temperature. Further, the 
.scales furnished by different gases are nearly identical since the volume 
and pressure coefficients of all permanent gases are nearly equal. 
Hence, gas thermometers arc used as primary standards with which 
-ill others are compared and calibrated. 

The theoretical bases underlying the use of gas as thermometric 
liiiii: :: are the laws of Boyle* and Charles which are very approxi- 
mately obeyed by the so-called permanent gases in nature but will be 
rigorously obeyed b] a perfect gas. Let a gas be initially at pressure 
/j,, volume V\ and temperature r t °C. TF we first change its pressure 
from p! to p^, keeping the temperature constant, and next change the 
temperature from t x to t 2 , keeping the pressure constant, we have 
n 'in these laws, 

. , ti' 1 + ox, 

where v' is the intermediate volume and a the coefficient of expansion 
at constant pressure, which is found experimentally to be approxi- 
mately equal to 1/273. Combining these two equations we have 



Pi v i 1 " ' i 



/V's 



l-ffcf« 



(1) 



which is the gas equation. 

II >'. [/a, ^2 = 0, U» at the temperature l/a = 273°C 

i li ice point, the volume of the gas would be zero 

provided the perfect gas equation is obeyed throughout the range, 

This temperature is, by definition, called the ah &ro. It is true 

ih: gas would liquefy and solidify long before this stage is reached 

i in I the perfect gas equation would cease to be valid. Further, it is 

inconceivable that matter should at any rime occupy no space (w=0). 

Nevertheless l of such a zero of temperature is very useful. f 

II we lure temperatures from this zero, the ice-point is given 

by I/a, the steam point by (l/«)-|-100, arid generally any tempera tun-', 

by (I /«) -H 2 =7YK, Th c scale so obtained is called the Kelvin 

• rale and will be denoted by °K, Hence equation (1) 

i m omes 



Pi"j 






n 



or 



^2 J l ' 



where the suffix denotes the quantities at 0°C. 



• ■ (2) 
The quantity 



■H Boyle (1627-1691) was b m in Trehnd but settled in England m 
He distinguished himsdi in the study of Physio and Chemistry and was 

"| i the foundation members of the Royal Society. TTis main contribution is 

I In' law of sases which still bears bis name. 

1 I he absolute zero thus defined is shown from thermodynamic considerations 
to be the lowest temperature possible. Hence the idea of this absolute 
rry important [see further §71. 






: iMOMETTRY 



[chap. 



. , T is known as the gas constant and varies as the mass of the 
gas taken, but is approximately the sump for equivalent gram-mole- 
cules cl all gases. For one gram-molecule this quantity is usually 
denoted by R and is equal to 8.3 X ^ J " ergs/degree approximately. 
If the mass of the gas is increased n times, the volume at the same 
temperature and pressure will he increased n times and hence the gas 
constant will also he increased n times. Hence, the gas equation cm 
he. written generally in the form pv=nRT where n denotes the num- 
ber of gram-molecules of the gas. 

Equation (2) furnishes two ways of measuring temperature. The 
pressure may be kept constant and' the volume observed at different 
temperatures giving us the constant pressure thermometer ; or the 
volume may be kept constant and the change in pressure noted, a 
principle utilised in the constant volume thermometer. . 

Exerctee. — Calculate the pressure of 20 grams of hydrogen inside 
a vessel of 1 cubic metre capacity at. the temperature of 27- C. 

[pv=?iRT where n=2Qf c Z. Ans. 0.25 atm.] 

5. Callendar Compensated Air-Thermometer. — Accurate measure- 
ments with the constant-pressure gas thermometers are difficult as 

the gas in the connecting tube and 
the manometer is at a tempera 
different from that of the bulb. To 
avoid this Callendar devised the 
compensated air thermometer. In 
tins instrument (Fig, 3) the pressure 
of the aii in the thermometer bulb B 
i the pressui e of the 
air in D as indicated by the sulphuric 
acid gauge G. When 11 is heated, 
the pressure of the air in B increases 
and equality of pressure is restored 
by allowiifg mercury to flow out from 
the mercury reservoir S. The volume of the tube connecting B and 
S is eliminated by attaching to D an exactly similar tube placed close 
to it. This will 'be evident from the following consideration: — 

Let v, v it tfg be the volume of the bulb B, the tube conned 
B and S P and the air in S respectively ; 6, lt a , their respective 
temperatures, n the number of gram-molecules of air contained in B, 
S and the connecting, tube anil p its pressure, we have from the 

gas laws, 




. 



*(t+i a-* 






Similarly if ?'', v/ be the volumes of the air in D and the tube attach- 
ed, 0', 0/ their respective temperatures, n f the number of gram- 
molecules and p' the pressure, we have 



'{r+i)-«- 



W 




STANDARD fiAS 1 MEIERS 

-«', * =*/, h -= &,', wc have from {3} and (4) 



,i dition S and D are immersed in melting ice, a =0' =firf, 

reezing , water, and 



ine v- tf, V 2 by B, D, S 



B 



= d °-n-s 



(5) 



Mi us we see that the influence of the connecting tubes is entirely 

i a bed if (I) the pressure in B is kept equal to that in D (p — ff)\ 

i he total mass of the gas in B, S, and the connecting tube is 

to that in D and the connecting tube ,{n = nf) ; (3) the volumes 

Or tl ; ecting tubes 

anal (v-i — vf). The 

i a nd i t i o il [di—8i) 

is automatically satisfied 

i e ilie two connect- 

tubes are placed 

side by side and are at 

i In- same temperature. 

6. Standard Gas 
Thermometers. — T h e 

i on slant-pressure a i r- 

thermometer has been 

nded by Callendar 

ii . arious grounds : (1) 
ih apparatus and the 
. ulations are simple ; 
i he internal pressure 
on (he bulb does not in- 
, as the tempera- 
ture vises; (3) accuracy 
of the results depends 
i ii the accuracy of 
hing. . tevertheless, 
instrument dees not 
give i : ■ i'. • rdant results 
1 has been replaced 
the constan t-volume 
thermometer ns a stan- 
l. The normal ther- 
icter selected by the 
Bute au 1 ntem a ti on al pig, 4 (a ) .— ■ Constant-volume Hydrogen Thermometer, 

Poids et Mesures 
and everywhere adopted today is the constant volume hydrogen tl\er- 




THERMOMETRY 



[CHAP. 



meter filled witli gas at a pressure of I metre of mercury at the 
temperature of melting ice. It consists essentially of two parts ; the 
bulb enclosing the invariable gaseous mass and the manometer for 
measuring the pressure. Fig. 4(a) represents the thermometer dia- 
gr annua tieally. 

The bulb C is a platinum-iridium tube a litre in capacity, 1 metre 
in length and 3G mm. in diameter. It is attached to the manometer 
by a capillary tube of platinum 1 metre in length. The manometer 
consists of two tubes A and B, and the stem of the barometer R dips 
into A. The barometer tube is bent so that the 
upper surface of mercury in it. is exactly above 
\i and these levels can be read oft by a catheto- 
meter furnished with telescopes. The number 
of observations to be taken is thus reduced to 
two. B consists of two columns of mercury 
separated by the sted-picce H and both these 
columns communicate with A. By raising or 
lowering the mercury reservoir M, the mercury 
surface in the lower' part of B is arranged just 
to touch a fine platinum point P [shown separ- 
ately in Fig. 4(6)], projecting from the steel- 
piece H, and thus the volume of the enclosed 
gas is kept, constant. 

The thermometer described above is suit- 

:: for measuring temperatures up to 600°G, 

iperatures certain modifications 

ary which will he discussed under 

netry' in Chapter XI. The range of gas thermometers with 

Ten irons ran be extended from -200° to 1600 "C. 
We shall now deduce a Eormula for converting the observed 
pressn nga into corresponding temperatures. T.I: /?, ; „ p vn denote 

the pressure indicated by the manometer at the ice point and steam- 
point respectively, tlien p„v = riRTn, pimV — nR(T -{- 100) where T 
represents the ice-point on the per Feet gas scale and the fundamental 
interval is 100°C. Hence 

to *v 

rr ioo /to p > 

where 8 is the coefficient of expansion at constant volume. Thus 
we know ft for that gas from a measurement of p w and /_v To find 
absolute temperature corresponding to any observed pressure p t , 
we have, 







Tx ~Po 



m 



PERFECT GAS SCALE J 

since T A = 1/8. Thus to determine an unknown temperature the 

corresponding pressure *, is observed on the thermometer and he 

temperature 1\ ^calculated either from the above relation or graphical y. 

In an actual measurement corrections have to be applied for the 

following : — 

(a) The gas in the 'dead space' is not raised to the ^.nperatiire 
of the bulb. The Mead space' consists of the space }»*" <^f 
tube and in the manometer between the mercury level and the steel- 
piece H. Its initial and final temperatures are also different. 

, Increase in the volume of the bulb C with rise of temperature. 

(c) Change in volume of the bulb due to changes in internal 

3S "d)' Changes intensity of mercury on account of temperature 
changes, 

For a discussion of these, the authors' book ',4 Treatise on Heat 
may be consulted. 

THERMODYNAMIC OR ABSOLUTE SCALE OF 
TEMPERATURE 
7 Perfect Gas Scale.— The formulae developed for the gas 

thermometer assume that the gas, in question, accurately obeys the 
E?as taws but experiments show that no real gas does so exact!), ine 
coefficient of expansion « at constant pressure is not exactly the same 
for veal eases as may be seen from any book of physical constants {see 
Rave and Laby : 'PIMical Constants'). Further the two coefficients 
* Fare not exactly equal, and also varies with the initial pressure 
Thus, different gases would furnish different scales of temperature ij 
the thermometer is calibrated as indicated above, and the selection 
of anv particular gas will be arbitrary and will give an arbitrary scale 
of temperature. To avoid this arbitrariness we must reduce our 
ovations to that state of the gas.in which the perfect gas* equation 
i, satisfied. 

We shall now indicate the mediodsf of reducing the observations 
on real gases to the perfect gas state. This can be done when we 
have knowledge of the deviation of gases from Boyle s law. 1 he 
calculations are rather complicated and will not be given here. It is 
enough to point out that the equation for any real gas can be written 
in die form 

PV. z=-.RT + Bp +\Cp* + D / ,a + ' 

where! B, C, Z>...aie constants which go on decreasing rapidly. Thus 
ii is evident (hat at infinitely low pressure (p-*0) all gases will obey 
Boyle's law accurately and this conclusion is borne out by expert 

* A perfect gas is defined as one which will obey Boyle's law and Joule's law 
(Chap. II. sec. 22) rigorously. 

t For fuller information sec A Treatise ott Heat by the Authors. 
J Further see Chap. IV, Section 1. 







10 



II!' I!. MOM IMR1 



[t KAP, 



rnent;. ation. Now it is mentally found that the tempe- 

rature scales, obtained by using the different gases and extrapolating 

observations to zero pressure, art: actually identical lor all the 
This is the perfect gas scale. The coefficient of expansion « 
for a perl: an be calculated in this manner from the data given 

by Heuse and Otto. The mean of several results gives a = 0.00 
for a perfect gas. Thus T the melting point of ice on the perfect 
gas scale— l/« = 273.10°K, Considering all the available data Birge 
adopts the value 273.16^0.01. We shall use the value £7:: 
or 273. pending upon the degree of accuracy required. Aunin 

i ;j accurate experimental data on oxygen we have 



Lim ( 



22.41-1 litre X atmospl 



Hence 



fl = Lim (pV)„ 



22.414 x 10 a X 76 X 13.595 X 081 ., 

97 * Tfi ergi/degree 



273.16 

= 8.314 X I(|T : 

1.186 X ' 'I' 11 ' 1 ' III), hence R=l 

, per degree. This is the value of the gas constant. For real 
alue of the quantity p G f',,/7',, differs only slightly From this, 
v point out i alt arrived at in Chaj 

tvn how Lord Kelvin, of heat- 

scale independent of the property 
scale of temperature 
the thermodynamic scale. Further it is shown there 
i uite identical with the perfect gas scale. We thus 
see that th< . scale which was hitherto shown to depend 

on the properti; is now becomes independent of the properties, 

any particular substance. Hence it is called absolute scale 
standard scale adopted in scientific work. 

r method of obtaining the correction to be applied to the 
real gas scale consists in performing the Joule-Thomson experiment 
(Chap. VI). But unfortunately the existii g data on Joule-Thomson 
effect are no tent to enable us to apply this method and the 

method given above is almost universally employed. 

S. Standardization of Secondary Thermometer*. — Gas the^ntome- 

• are very cumbersome to use and require several corrcctio 

Hence in labor a to are replaced b; secondary standards, such 

as the resistance thermometer, the thermocouple, etc., which have 

been carefully standardized by comparison with a standard gas thermo- 

er in standardizing laboratories like the National Physical Tab. 

or the Bureau, of Standards in Wa 



RE BATHS 



I I 



pur] imparison baths may be constructed, each suitable foi 

particular range. Between 0° and* 100°C a water bath, between 80° 
and 250°C an oil bath, between 250° and 600 q C a mixture of potas- 
sium nitrate and sodium nitrate., and above that an electrical heater is 
generally employed, 

The secondary thermometers may also be standardized by means 
of a series of easily reproducible fixed points whose temperatures have 
been accurately determined. A table ot standard temperatures is given 
below (Table I.) . The values are generally those adopted by the 

r enth General Conference of Weights and Measures represi 
thirty-one natrons which was held in October 1027, but some amend- 
ment-, made by the Ninth General Conference in 1948 have also been 
incorporated.* 

9, Fixed Temperature Baths. — It is frequently convenient to 
calibrate the secondary thermometers by means of the fixed-point 
scale given in Table 1. The ice-point may be most conveniently 
obtained by dipping the thermometer in pure melting- ice contained in 
a dewar flask. This is a double-walled glass or metal vessel whose 
sides are silvered.-]* For the steam-point the hypsometer indicated 
in Fig. , r >, p. 12 is employed. The diagram explains itself. C is the 



v I, — Standard Temperatures. 



Temper?' i 


Substance 


Temperature 


Substance 


Centigrade 




tvti nade 




-252.780° 


E. P. of Hydrogen 


419*5 


F. p. of : 


-195 


B. P. 


444 -vi 


B. P. of Sulphur 


- 182 


:. P. of Oxygen 


630 -5 


F. P. o : Anil :i 


- 7S-5 


Sublimation of CO» 


9m -8 


M. P g Ivor 


- 38-87 


F. P. of Mercury 


1063 


M. P. 


0-00 


:■,:. p. of ke 




F. P. of Coppi 


+ 32-38 


Transition temperature 


1453 


U. P. ol Nickel 




of Na.SOaOH.0 


1553 


F. P. of Palladium 


ioo-orjo 


B. P. of Water 


1769 


F. P. of Platinum 


218-0 


B, I ■ > hLbalene 


2443 


F, P. of Iridium 


231-9 


R P. of Tin 


2620±10 


M. P. of Molyb- 


305-9 


B. P. of Bcnzophenone 




denum 


320-9 


F, P. of Cadmium 


3380±50 


M. P of 


327-3 


F. P. of Lead 


3500+50 


-M. P. of Carbon 



*See "The International Temperature Scale of 1948", National 
Laboratory, Teddington (1949), 

tFor a complete description see Chap. VI. 



32 



THERMO M KTRY 



[chap, 



condenser employed lo present the water from being lost by evapora- 
tion, M is the manometer and T the thermometer. The path of steam 
is indicated by arrows. The boiling point of water at the pressure p 
(in mm. of mercury) h found to be given by the relation 

* = 100.000-1-S.67 X I0~ 2 (p - 760) - 2.3 >< 10-* (p - 760)2. 

For other fixed points a number o£ vapour baths in which sulphur, 

naphthalene, aniline etc. are used, serves the purpose, 

_ For determining the boiling point of sulphur Callcndar and 
Griffiths found that the standard Meyer-tube apparatus was verv 
suitable. It consists of a hard glass cylinder A of diameter 5 cm. 
and length about 25 cm. to which a spherical bulb B is attached at 
the bottom (Fig, 6) , The whole is surrounded by an asbestos chamber 
C. The thermometer T is fitted with an asbestos or aluminium cone 





Fig. 5, — Hypsometer. 



6. — Sulphur-boiling apparatus 



D. This cone serves in Lwo ways : (1) it prevents the condensed 
sulphur from running down over the bulb and cooling it below the 
temperature of sulphur vapour ; (2) it prevents the bulb from directly 
radiating to the cooler parts of the tube. Sulphur is placed in the 
bulb and heated aver a flame. A side tube may be provided in the 
upper part of the chamber and serves to condense sulphur vapour. The 
boiling point of sulphur is given by the formula 

t = 444.60+9.09 X 1«~ 8 (P -760)- 4.8 X 10 •"(/>- 760)*. 




PLATINUM THERMOMK'iKKS 



13 



C|P c 



Baths for naphthalene and aniline may be constructed by slightly 
modifying the above apparatus. 

RESISTANCE THERMOMETRY 
10. Platinum Thermometers.— The necessity of 
, ndary standards has been clearly indicated above. 
:> tvpes of such instruments based on two electrical 
properties of matter will be described in this chapter. 
These properties are— (1) variation of electrical resis- 
tance of metals with temperature; (2) variation of 
thermal electro-motive force with temperature. First 
let us consider the former. 

Sir William Siemens was the first to construct 
ermometer in 1871 based on this principle but^he con- 
structional details were unsatisfactory. Later improve- 
ments by Callendar* and Griffiths have given the instru- 
ment its modem form. Fig. 7 represents an hermetic- 
ally-sealed thermometer designed by Dr. E. H. Griffiths 
for laboratory work of high precision. Pure platinum 
ire free from silicon, carbon., tin and other impurities is 
: ted. It is doubled on itself to avoid induction effects 
and then wound on a thin plate of insulating mica m. 
The ends of this wire are attached to platinum leads 
which pass through holes in mica sheets closely fitting 
the upper part of the tube, and the other ends of these 
ads ire joined to terminals P, P at the top of the instru- 
ct. The mica sheets give the best insulation and 
vent convection current of air up and down the tube, 
L e coil is sealed for, otherwise, moisture would deposit 
'in the mica and break down the insulation. To cpm- 
iisate for the resistance of the leads, an exactly similar 
pair of leads, with their low ends joined togeth 
placed close to the platinum thermometer leads, and is 
, umected to terminals marked C, C, on the instruir, 
hese are called compensating leads and are joined in 
the third arm of the Whcatstone bridge as shown in 
r, 8 (p. lft). Then since the ratio arms are kept equal 
;uid the compensating and platinum leads have equal 
resistance at all temperatures, it is the resistance of the 
inum coil alone which is determined. For work up 
to 700 °C copper leads may be used and the whole may 
l e : nclosed in a Lube of hard glass. But for high tem- Fjg 7^pi ati . 
peratures platinum leads must he used and the whole num. therm 
be enclosed in a tube of glazed porcelain. meter. 

The precision and reliability of modern resistance thermometers 
ate entirely due to the work of Callendar and Griffiths. They deter- 

* H. L. Callcndar (1863-1930) was educated at Cambridge and worked it; the 
I •, tidtsh Laboratory from ISSS to 1890 on resistance thermometry. His greatest 
are the development of the platinum resistance thermometer and the 
Investigation of the properties of steam. 



M 



MOMETKV 






MEASUREMENT Ol? RESISTANCE 



. 






latmum from 0° to 500*C and found 
'" ,L Itwas ™*y a ^n fay a parabolic formula of the 

= J2 «( 1 B*)* ■ . . . (7) 

C and 0»C and . 
, ,-,..9.J lO-» f = _$8 •- I0 : ■ it ,, i 
' ' li « i -.. Srytosoh q B u U a 

£c equation to find | from the value ol [ e „ ave ' 

*™ tothl ! lario ,P™ ducednomeE 

1,111 PWrt ";"• Thus we define the platn 
ans of the simple linear relation* Kinpen , t p by 

m 

''-fc=S ,o ° 

US' SiS 

ill 00/ 'loo' • • 






*r-i? 






100 



100a -(- (100) 3 /? 
_ £(100)* r/ mi i ] 
a+ 100^ H 100/ 
Thus S in equation (9) is equal to 



x loo |> / 



- !0Q£* 
tie specimens emploved is nhnm i r, T t, 

•« I ' Ufa different from n . The exact relation may he fcduc, 



4 

finding the platinum temperature ^ lor the boiling point of sulphur 
temperature is known, and then substituting in ( 
Kir i tor a specimen, use my unkno 

ature. The the thermometer at the unkno 

temperature is found out and ; ( tf ) h ** determined Froi 

this, using the value of 8, the true temperature i can he determined 
will jip of (9). Il will be ol that the correction i 

giving the value of t-t t ins the unknown temperature t. 

voik the value ol t f may be substituted for t on the right-hand 
(9) . Tor accurate work, however, the procedure is as follows : — 
Tin hand side of (9) is calculated for different assumed values 

ol I able is constructed giving the value of this correction t< I 

different values of tp. With the help of this table the true tem- 
perature i corresponding to the experimentally determined platinum 
tempera i in r t p is found. 

It was shown later by Heyoock and Neville, and Waidner and 

Bin i if die platinum thermometer is standardized at Q", 100° 

boiling point o( sulphur the parabolic formula (9) gives true 

L! Ear as 630 °C 

We shall illustrate the method by, 'a numerical example. Let 

the resistance ol a given platinum tliermometer at 0°, 100° and 

i 4.6) be 2*56, 8. 56 and 6.78 ohms resj 
equired to calculate the true temperature when the 
resis ' E the. thei mometer is 5.56 ohi 

U-- 5 ,' 5 . e X 100 = 3GG°C. 



t, 



"~ 3-5G-2-56 
for the boiling point of sulphur 




_ 6-78-. 
" 3-56- 

444-6-422 =S 



7,- * 



100 ^- 422°C. 



1 - 
1 1 id / 100 

hence & = 1.4. 

From (9) we get / = 300, (-*,== S.I-; i p = 291.2°, 

t — 320, t-t p = 10.4 hence f. = 309.6°. 
►rrection t-t p for t f = 291.2° is 8.8 and for 309.6 it is 10.4. 

: 300 it is B.B + ~=^-X (300-291.2) = 9,6. 

rherefore the true temperature t is equal to 309.6 

11. Measurement of Resistance. — The determination or tempera- 
. this thermometei Involves the accurate measurement ol the 
ace of the platinum wire. Various special types of resistance 
re used lor this purpose. In order to compensate for the 
ice ol the leads, a bridge with equal ratio arms i- used. Fui 
the brii g< should be capable of measuring cbaxi e to a 

ree of accuracy & fundamental interval is 

> and measurement of temperature to hundedths of a tl 
requi stance measurement to one ten-thousandth of an ohm. 






16 



THitRMOM I-TRY 



CHAP. 




Fig'. 8j— Calliflniaratid Griffiths Bridge. 

adjusted for no deflection of the galvanometer G. 



Then 



The Callendar and Griffiths bridge* is quite suitable for this pur- 
pose. Fig, 8 indicates r.he connections. Q and Si the ratio anas, 
arc kept equal by the makers of the instrument. R consists of a set 

of resistances of 1, 2, 4, 8,' 16, 32, 
64 units. The usual plug contacts 
are here replaced by mercury cup 
contacts. L x and L-. axe two paral- 
lel wires of the same material 
which can be connected to each 
other by the contact-maker K, 
This arrangement is adopted in 
order to eliminate thermo-elec.tr o- 
motive forces. P represents the 
thermometer and C the compen- 
sating leads. The resistance r 
acts as a shunt and makes the 
resistance of the wire exactly in 
the desired ratio. The bridge is 
Suppose the 
balance point is obtained with the key K at a distance x from the 
centre of the wire and die entire length of the wire is 2a. 

or P =r.R — 2x P . 

where p is the resistance per unit length of the wire. It is found 

it to select the wire L 3 and che shunt r in such a mannei that 

1 cm. o£ the bridge wire has a resistance of 1/200 ohm and the total 

of the wire is 20 cm., while the smallest resistance in R 

d 1 has a r: >. d stance of tfie whole 

i.e.,, 20/200 — 0.1 ohm. Thus if the fundamental interval of the 

it'Kc thermometer is 1 ohm the temperature can be determined 

C provided the balai at is determined correct to 

Oil mm. For accurate work., however, various precautions are neces- 

which are given belowj- ; — 

The current, flowing through the bridge heats the bridge 
coils and changes their resistance. The change in tem- 
perature may be observed on a thermometer and the 
corresponding change in resistance calculated. The correc- 
tion can then be easily applied. Or the bridge ni;i 
placed in a thermostat. 

The thermometer coil has to be very thin (0.15 mm. dia- 
:,Y) since it must have a large resistance and hence 
the heating effect is considerable. From Calendar's 
observations the heating effect for a current of .01 ampere 
is 0.016° at 0°C and 0.017° at 100 C C. According to him 
the best, procedure is to pass the same current through 

* For further details see Flint and W'or-v.o-j 'I'mciiaii I'kys'u-.-;'. 
i For full details see Methods of Measuring Temperature by E, Griffiths, 
(Chap. 3.) 



(2) 







B i 1-: A Sl.I REMENT OF RESIST A - 



17 



the thermometer at all temperatures when the heating 
effect remains approximately constant. 

(#) The bridge centre must be determined and the bridge wire 
calibrated. 

(4) Due to temperature gradient along the conducting leads 
and the junctions thermo-electromotive forces are developed 
in the circuit whose magnitude may be found by closing 
the galvanometer circuit when the battery circuit is kept 
open. To eliminate these the galvanometer circuit should 
be permanently closed and balance obtained for reversals 
of the battery current. If induction effects are perceptible 
when the battery * circuit is made or broken a thermo- 
electric key should be employed. This key first breaks the 
galvanometer circuit, then 'makes the battery and the 
galvanometer circuits in succession. 

(5) The external leads connecting the terminals PP, CO to the 
bridge should be exactly similar and similarly placed. 

12. As already mentioned the platinum thermometer is standar- 
dized by measuring the resistance at the melting point of ice, boiling 
points of water and sulphur. The last gives S and hence t can be 
determined from any subsequent determination of t p . Eqn. (9) how- 
ever does not hold much above 630°C and therefore every such thermo- 
meter is provided with a calibration curve drawn by an actual com- 
parison with a standard gas thermometer in standardizing laboratories. 
The temperature can be directly read from this tempera We-resistim ,e 
curve, 

The great advantage of platinum thermometers lies in their wide- 
range (~20Q°C to 121KPC). If carefully prepared, their readings are 
reliable to 0.01° up to 500°C and to 0.1° up to 1200°G but generally 
it is not desirable to use them above 1000°C owing to the dan- : 
contamination toy the insulating materials. They are free from 
changes of zero for the wire when pure and well-annealed has always 
the same resistance at the same temperature. They are very con- 
venient for ordinary use n.nd, when onee .standardized by comparison 
■ ith a gas thermometer, they serve as reliable standards. They are 
employed to measure' small differences of tempera tit re very 
ately, sometimes even to one ten-thousandth of a degree. There 
re, however, some drawbacks also. The resistance thermometer 
has a J: tge thermal capacity and the covering sheath has a low thermal 
Conductivity and therefore the thermometer does not quickly attain 
ilt:- temperature of the bath, in which it is immersed. Further some 
i ! Lost in balancing the bridge. For these reasons the resistance 
lometer is useless for measuring rapidly-changing temperatures. 
urther impurities in the platinum do not obey the same resistanre- 
I mperature law as the pure metal. 

I 2 gives the variation in resistance of a platinum therm o- 
metei over a wide range. It is taken from Henning's Temj 

. The value of the quantity R = R t /R is given for various 
o 



i;-, 



THERMOMETRY 



[chap. 



temperatures where H„ R are the resistances at tempera tine-:. 
m! ii ( ! respectively. 

Table 2,— Values of R—R t /R^ 



Ten:- . 


R 


Temp, 


R 


Ten 


J? 


D C 


021607 


°C 








-m 


+40 


1.15796 


280 


2.06661 


-180 


0.25927 


60 


1.23624 


300 


2.13931 


-160 


0.34505 


80 


1,31406 


320 


2.21154 


-140 


0.42986 


100 


1.39141 


340 


2J28330 


-120 


0.51347 


120 


146830 


360 


2.35460 


-100 


O.S9i 


140 


1.54471 




2.42543 


- 80 


0.67814 


160 


1.621 


400 


2.49S80 


- 60 


0.7594$ 


180 


1.69616 


420 


.570 


- 40 




200 


1.77118 


440 


2,63513 


-J 20 


0,92033 


220 


1.85474 


•li-:i 


2.70410 





1.00000 


240 


1.91983 


480 


2. 77261 


20 


1.07921 


260 


1.99345 







THERMO-ELECTRIC THERMOMETRY 

13. Thermo-Couples. — Let us now return to the second electri- 

asurement. Starting from 

■ ., in 1821 nm ■• were made to 

ometer based on this principle, Eor instani 

Pouille( and Regnault. At present thermo-electric 

attained a degree of precision inferior only to 

rice thermometry below 1000°C, but foi atrires exceed- 

::ic it [g the onl) sensit: I convenient electrical method 

at our disposal. 

A thermo-electric thermometer installation consists of the follow- 
ing parts :— 

;]; The two elements constituting the thermo-couple. 
' The electrical insulation of these wires and the protecting; 
tubes. 

(3) Millivoltmeter or potentiometer for measui in- the thermo- 

electromoi h e force, 

(4) Arrangement for controlling the cold-junction temperature. 

The choice of the elements constituting the couple is determined 

tperature to which the couple is to be heated and .the e.m.f. 

developed. For low temperatures up to 800°C couples of base mewls 

such as iron-eon sf a ntnn and copper-constantan are satisfactory, as they 

lop a large e.m.f. of about. 40 to fif) microvolts per degree. For 

H atures these base metals cannot be used as thev get 

ized and melt. Nickel-iron couple may be used up to 600° while 

nickel-nichrome and chromel-alume) thermo-couples can be used up 



'-] 



THERMO-GOUP LES 



19 



to 1000° but above that platinum and an alloy of platinum with 
iridium or rhodium must be used. Le Chatetier in 1886 introduced 
the couple consisting of pure platinum and an alloy of 
90 per cent Pi and 10 per cent Rh which is no 
employed for scientific work. The e.m.f. developed by 
these noble metals is, however, much less. 

The two elements are taken in die form of a wire 
aud one end of both is welded together electrically or 
in an ox y hydrogen flame. This end « (Fig, 9) forms 
the hot junction. The portions of the wires near the 
hot junction are insulated with capillaries of fire- 
clay (or hard glass for lower temperatures) and are 
threaded through mica discs enclosed in outer protect- 
ing tube of porcelain, quartz or hard glass, depending 
upon the temperature for which it is meant. The 
protecting tube prevents the junction from contamina- 
tion but necessarily introduces a lag. For rough use 
this may be further enclosed in a steel sheath (shown 
black in the figure) . Where there is no risk of conta- 
mination, the mica discs and the protecting tubes can 
be dispensed with. The wires are connected to terminals 
d ana! C s on the instrument. To these terminals are 
connected flexible compensating leads leading to the 
cold junction ' (Fig. 10a) . These leads are usually of 
the same material as the elements of the couple itself. 
Thus the cold junction is transferred to a convenient 
1 1 hi mt place where a constant temperature, say Q C, 
can be maintained. Usually the compensating leads 
Hi; marked so that there is no difficulty in connecting 
jo the proper terminals. 

There are two ways of making the connections 
which are indicated in Figs. 10 (b) and (c) . The dia- 
grams explain themselves. The cold junction is 
immersed in ice at 0°C. As a recording instrument 

r a millivoltmeter or a potentiometer is employed. 



-a 

Fig. 9 — 
Thermo- 
couple. 



i i couple 



Cold 

JtUlCtiun 




Hat 

Illl'i til I- 



Cotnpeasating 
extension 




Hot 
junction junction junction 



W (6) 

Fig, 10.— Measuring with a tincnno-couple. 



:v: 



14. To Unci the temperature of the hot junction we must 

tire the e.m.f, &< bei « een thi enda ' : me o ►pper leads. 



20 



THERMOMETRY 



[CHAP. 



I li : s ran be done by means of a high resistance miilivoltmeter which 
mav be graduated to mperatures directly and die temperature 

thus obtained can be relied 
upon to about ±$°C. For 
attaining higher accuracy 





Fig. H.— Illustration of the principle 
of potentiometer. 



a potentiometer must be 
used. This arrangement 
essentially consists of a 
number of resistance coils 
A (Fig. 11) placed in series 
with a long wire resistance 
r stretched along a scale. 
A current from the battery 
E flows through these re- 
sistances and its sti: 
is so adjusted by varying 
R that the potential differ- 
ence across a fixed ■ 



ance X balances against the c.mX of a standard cadmium cell C 
{L0183 volts) . The e.m.f. developed by the thermo-couple Th is 
balanced as indicated. The potentiometer can be made direct reading 
bv keeping K=z 10 LP ohms. Thus there is a fall of 1 volt pe 
ohms and by constructing; the smallest resistance coil of fl.l ohm 
mce and the wire r also of the same resistance, the total e.m.f, 
: the wire will be 1 m.v. If the wire is divided into 100 d.iv 
and in addition lias a sliding vernier having TO divisions the readings 
)e taken correct to 1 microvolt. 

Various types of potentiometers based on this principle have been 
devised specially lor this purpose* With these insiniments the 
e,m.f. can be measured accurately to 1 microvolt which corresponds 
!,, r a pt— Pt-Uh couple. For a copper-constantan couple 
this corresponds to about 1/40 degree. With a sensitive arrangement 
it. is possible to measure to 0,1 microvolt when the sensitiveness is 
increased about ten times. For accurate work the cold junction must 
be maintained at 0°C otherwise corrections! will be required in that 
respect. 

In order to deduce the temperature from an experimental deter- 
mination or the e.m.f. a calibration curve is generally^ supplied with 
the instrument. This gives the temperature corresponding to different 
electromotive forces developed and has been drawn by the makers 
by an actual comparison with a standard thermo-couple Throughout 
the range. If it is required to calibrate a thermo-couple in the 
absence of a standard one, the fixed points (Sec, 8) must be utilised. 
The e.m.f. at those points is measured and an empirical interpolation 
formula employed in order to give the e.m.f, corresponding to the 

* A description of these will be found in Methods of Measuring Temperature, 
by E, Griffiths. 

f See Ezcr Griffiths, Methods of Measuring Temperature, (1947), p. 74. 




THERMO-COUl'LhS 



21 



rmediate temperatures. For a Pt— Pt-Rh couple three different 
equ: must be used for the different ranges. Thus 

; am 0° to 400*0, E = At + &(l- O , 
300° to 1200 C C, E = - A* + & + &?, 

1100° to 1750<>C, E = -A" + B"t -f- C"l* 
where A, B, C are constants whose values are empirically determined. 
Thermo-couples are frequently employed for laboratory work since 
v are cheap and can be easily constructed. They can be used for 
i measurement of rapidly-changing temperatures since the thermal 
-city of the junction is small ami hence the thermometer has 
. ctieally no lag. Another advantage in the use of thermo-couples is 
they' measure the temperature at a point—the point at which 
the two metals make electrical contact. Its chief disadvantage lies 
the fact that there is no theoretical formula which can be extra- 
polated over a wide range and consequently every thermo-couple 
requires separate calibration. 

The useful range of thermo-electric thermometers is about - 200 & 
to 1600°C. Readings are reliable only, when the composition of the 
i ile does not change even slightly. In actual practice frequent 
calibration is necessary. 

The following are the chief sources of error in thermo-electric 
i mometry : — 

(1) Parasitic electromotive forces developed in the circuit. They 
are due to (a) Peltier effect or e.m.f, developed due to heating of 

. t.i.ort of dissimilar metals at points of the circuit other than the 
i and the cold junctions. This occurs often in the measuring 
>aratus ; (b) Becqucrel effect or e.m.f, generated due to inhomo- 
geneities in a single wire; this occurs mainly in the thermo-couple 
wires. The e.m.L measured is a sum of these quantities and the 
Peltier e.m.f* at the two junctions and the Thomson e.m.f, along 
homogeneous wires of the thermo-couple with ends at the two tempe- 
ratures. The undesirable effects mentioned in (a) and (b) must be 
dilated by the use of materials and methods free from these effects 
nee they are not taken into account in any thermo-electric formula?. 

(2) Leakage from the light mains or furnace circuit. If leakage 

s through the potentiometer their presence can be detected 
1 1', short-circuiting the thermo-couple when the galvanometer continues 
to be deflected. 

I) Cold-junction correction if it is not kept at (PC. 

For the methods of minimising or eliminating these errors the 
is referred to Measurement of High T wres by Le 

Chatelier and Burgess. 

Table 3 compiled from various sources gives the thermo-electric 

J. for various couples in common use. The cold junction is 

• i, -lined at 0°G* and the hot junction at t°Q. The e.m.f. of the 

'•no-couple AB being positive means that the current flows from 

\ to P. at the cold junction. 



22 THERMOMETRY [CHAP. 

Table 3. — E.m.f. in Millivolts for some thermo-couples. 



Temp. 


90Pt-10Rh 


.'Ni 


Ag/Pl 


Fe/cons- 


Cu/cons- 


fC 


against Ft 




- 


tarrtan 


tantan 


200 


-S.27 


--. 


-100 








■ 


3.349 


- 80 




— 1.68 


-0.30 









r;.co 


0.00 


;:i::i 


0.00 


0.000 


+ 1 


+0.643 


+5.18 


+0.72 


I 5.40 


+4.276 


200 


1.436 


-.96 


+1.73 


10.99 


9.285 


m 


2.315 


752 


2.96 


16.56 


14:' 


400 


3. 250 


9.83 


4.47 






500 


4,219 


12.04 


6.26 


27,58 




l.llii 


5.222 


1450 


8.25 






700 


S.260 


17.30 




39.30 




800 


7.330 


20.73 


13,17 


45.72 




000 


8.43 


24.19 


15.99 








9.569 






. 






US 












14.312 










1600 


16,674 












17 Ml 











15. in methods of measuring temperature utilise the radia- 

; to he measured. 

will be discussed in detail later (sec Chap. XI). "J 
I for measuring temperatures from about lOOO^C to any 

upper limit. 

16. Certain other methods of measuring temperature utfljse any 
one of the following properties of matter : — 

(1) Expansion of a bar of metal. 

(2) Changes in vapour density with rise of temperature. 

(3) Variation of refractive index oF a gas with temperature 

accordance with Gladstone and Dale's law. 
": Calorimetric methods based on the measurement of quantity 
oi h 

: Change of vapour pressure with temperature. 

17. Low Temperature Thermometry.* — The standard thermo- 
meter in this range is the constant volume hydrogen or helium thermo- 
meter. The difficulty in this case is that gases liquefy and even 

lify at these low temperatures. Prof. Dewar, however, showed 
that the boiling point of hydrogen as indicated by the liydrog 
thermometer was -253.0 o C and -253.4°C, while a helium thermc 
meter registered -253.7°C and -252.1°C, Similarly he compared 

* For pyromelry see Chap. XT. 



LOW TEMVl'UATURE THERMOMETRY 



23 




thermometers of other gases. These experiments evidently led to 
the conclusion that a gas could be relied upon almost to its boiling 
point. Thus helium furnishes the scale down to its boiling point 
(4.2° K.) . The corrections necessary to convert this scale to the 
thermodynamic scale may be obtained and have been given by Onnes 
and Gath. For temperatures below 4.2°K we must use the helium 
gas thermometer with die pressure well below the vapour pressure 
of liquid helium at the temperature to be measured, so that the gas 
will not liquefy at that temperature. With this device the helium gas 
thermometer gives us the thermodynamic scale down to 1°K. Now 
we shall consider the secondary standards. 

Mercury freezes at -38.87 and alcohol at -1]1.8'"C and hence 
these thermometers cannot be used below the respective tempera- 
tures. A special liquid thermometer containing fractionally distilled 
petroleum ether can be used down to - 190*0, 

But for all accurate work, however, resistance thermometers are 
employed. It is absolutely essential that the substance of which the 
thermometer is made is perfectly pure. Pure metals show a regular 
rise in resistance with decrease of temperature. Dewar and 
Fleming found that the presence of the slightest trace of impurity in a 
metal is sufficient to produce a considerable increase in resistance at 
these low temperatures. It is, therefore, difficult to trust the purity 
of any specimen for very low temperatures without actual comparison. 
Henning found from a detailed investigation that the parabolic formula 
did not hold below — 40 D C. Van Dusen proposed the formula 

where e is a constant and 8 has already been defined on p. 14. The 
constants, R lf! K im and 8 are determined by calibration at 0*0, 
mil c; and the boiling point of sulphur as explained previously, and 
the constant e is then determined by calibration at the boiling point 
of oxygen (- I82.97°Cj . Van Dusen s formula has been found to hold 

atisfactorily from 0°C to -190°C, die error nowhere being greater 

ban ri=0,05 a . 

For temperatures lower than - 190°C, the platinum thermometer 
was used by Henning and Otto, and can be used with advantage up to 
20°K. There is, however, no satisfactory, formula for calculating the 
temperature from the observed resistance and a calibration curve has 
to be used. Sometimes lead and gold thermometers are also employed. 
Onnes has used lead down to -259°C and Nernsl lias, given a method 
dculating these temperatures. Below —250*0 resistance ther- 
mometers of cons tan tan and phosphor-bronze have been employed, the 
latter being much more sensitive. 

For low temperatures copper-cons tan tan and iron-cons tantan 
couples are very sensitive as they develop a large e.m.f. They can 
be used down to - 255 B C. They must be calibrated by direct' com- 
Eson with a gas thermometer. 



^4 



THERMOMETRY 









In order to measure temperatures below the temperature ol 
helium (-268°C) the vapour-pressure thermometer o£ helium can be 
employed, Its use is based on the well-known fact that the vapour 
pressure of a liquid varies uniquely with the temperature. Thus the 
method consists in measuring the vapour pressure of a liquid at the 
required temperature by means of an apparatus similar to that shown 
in Tig. 4, Chap. V, and obtaining the corresponding temperature by 
means of a calibration curve or a theoretical formula. The helium gas 
thermometer and the vapour-pressure thermometer have been used 
down to about 0.75 C K. For measuring still lower temperatures the 
paramagnetic susceptibility of salts is utilised. 

18* International Temperature Seal*.— We have seen that the 

thermodynamic centigrade scale is the standard scale of temperature 

d is given by the helium gas thermometer, but gas thermomel 
involves many experimental difficulties. On account of these difficulties 
in the practical realisation of the thermodynamic scale the Inter- 
M iiovial Commitee in 1927 found it expedient to adopt a practical 
scale known as the International Temperature Scale. This scale agrees 
with the thermodynamic scale as closely as our present knowledge 
permits and is at the same time designed to be easily and. accurately 
reproducible. It is based upon a number of reproducible fixed points 
to which numerical values have been assigned and the intermediate 
temperatures have been defined by agreement as the values given 
by trie following thermometers according to the scheme given below ; — 

(Pi 9 1 and and platinum resistance 

neter calibrated at D , I00°C and the boiling point 
sulphur. 

: From -190 platinum resistance thermometer 

which gives temperature by means of the formula 

Et = R (I -f- <xt + fit 2 + y (« - 100) **}, 

the four constants being determined by calibration at ice, 
steam, sulphur and oxygen points. It, will be seen that i.iv- 
formula is equivalent to (10) where e s= (I00) 2 87//J. 

(3) From 661>°C to I0b"S°C.— The platinum Pt-Rh thermocouple 

where temperature is defined by 

E = a -f bt 4- cF r 

and the three constants are determined by calibration at the 
dug point of antimony and at the silver and gold points. 

(4) Above 1063°C— An optical pyrometer (see Chap. XI) cali- 

brated at the gold point (106S D C) . 

it should be emphasized that the International Scale does not replace 
the thermodynamic scale ; it merely serves to represent it in a practical 
ter with sufficient, accuracy for most purposes. 



>•] 









0«K- 

lOO'K- 

273-K- 



IflO'C- 




200 



300 



m 



SCO 



6Q0 



TOD — 



SOD — 



: :o 



:;. :: 



CHART OF FIXED POINTS 



Ch art of Fixed Po ints 

-iS-iOLi'. B. P. of Helium- 
-283.-7E 1 B, V. of HydrDgeu 

-LS5'6J B. P. of Nitrogen 
^Ug-S«a B- P. of Oxygen 



- 78-4E3 Sublimation of CO* 

- ss-632 F. P. of Ifenury 

D'OU Water freaes 

• 9S'3S4 TroBsSttonofNa^lOHjO. 

■ lOft-Ott B, P. of Water 



2?) 



419 '43 
4,14 '60 



SITS* B. P. of. Naphthalene 
231*86 P.P. of Tin 



305-90 B. P. of Bensoptienmic 
320-9 F . P. of Cadmium 



1 r - 

B. P. of Sulphur 



' .•■ .:, ' . . 



523*C. Draper point 
iBed Just 

Visible 



SKI'S F . P. of Antimony 
600' I F. E of Alurnjniym 



m-9, F, P. of Silver 



Dull red 



Cherry 
red 



Bright 

□harry 



1 



& 



■-; 



-' 






26 



i -.-"-I 






i 



I'-I.M 



liO'J 






!':'•' 



2400 



SWO 









THFRMOMKTKY 

Chart of Fixed Points (Contd}* 



[chap. 



M. P. of Gold 
p.P.ofCopJn:r 



■1*53 F.P. ofNkNri 
■15S2 F.P of Palladium 

176& F. P. of Platinum 



F.P. of Iridium 
P. pf Molybdenum 






MOO 



H 







40MT- 



■ 3850 M, P, of Timgaten 



Dun 
Oxaage 

Bjiebt 
Orange 



Whit? 



Dasslme 



1 

I 



M 



ILLUSTRATION OF PRINCIPLES OF THERMOMETRY 

19. Illustration of the Principles of Thermometry, 

Absolute or Thermodynamic Scale 

Gas Thermometers 
(Primary standards) 



27 



Constant Volume 



Constant Pressure 



Gravimetric 

Thermometers 
(Compensated 

Air thermometer) 



'Secondary Thermometers 



r — r T" ~i 

Expansion "Resistance Thermoelectric Radiation 

Thermometers Thermometers Thermometers Pyrometers 



Bo oka R ecom men (led 

1. Burgess and T.e Chatelier, The Measurement of High 

Temperatures. 

2. Ezer Griffiths, Methods of Measuring Temperature, (Griffin, 

1947). 

■. Hcrniing, Temperaturmessung. 

4, A Dictionary of Applied Physics (Glazebrook), Vol. J, Article 

on Thermometry. 

5, J. A, Hall, Fundamentals of Thermometry, Institute of 

Physics, London (1953). 

6, J. A. Hall. Practical Thermometry, Institute of Physics, 

London (3953). 



Other References-. 

Temperature, its Measurement and Control in Science and 
Industry (1941) , published by Reinhold Publishing Corpo- 
ration,, New York. 







METHOD OK MIXTURES 



24 



CHAPTER II 
CAL0R1METRY 



1. Quantity of Heat. — It is a matter of common experience that 
when a hot body is placed in contact with a cold one, the former 
becomes colder and the latter warmer ; we say that a certain quantity . 
of heat has passed from the hot body to die cold one. But a sim| 
experiment shows that when different, bodies are raised to the same 
temperature and then allowed to exchange heat with a cold body the 
final temperature is different. If we take equal quantities of water 
in three different vessels at the same temperature and plunge equal 
masses of aluminium, lead and copper previously heated to JOO'C into 
diese vessels, one in each, the equilibrium temperature is highest for 
aluminium and least for lead. This indicates that, of these three 
metals, aluminium can yield the largest quantity of heat and lead 
the le 

For measuring quantities ol: heat we require a ' unit/ The 
quantity of heat required to raise the temperature of 1 gram of 
water through 1°C is called the 'calorie' which is also the thermal 
unit tot measuring quantities of heat. The 15°C calorie is defined 
as the quantity of heat, which would raise the temperature of oi 
gram of C to 15.5°C and has been re ided hv 

the lied Physics (1934) for 

andard. In Britain the British thermal unit 
tly employed* which represents the 
qua heat required to raise 1 lb. of water through l°I\ 1 

of any substance is defined as the number of calories 

. u of the substance through 1°C. Tin's is 

peaking not the same at all temperatures. Thus if a quantity 

. the temperature of m grams of a substance from b 

to 8% s f the mean specific heat of the substance, is given by 

| m {B'~9) ] ; while if a quantity dQ raises the temperature by d&* 

tin specific heat at the temperature $ is —^ , 

The thermal capacity or water equivalent of a particular body 
he product, of its mass and specific heat. 

2. Methods in Calorimetry.* — 'The following are the chief methods 
iloyed in Calorimetryf : — 

(1) Method of Mixtures. 

(2) Method of Cooling. 

* Sometimes the lb. calorie or cen%rade heat unit ("C. H. U.) or centigrade 
thermal unit (C. Th, U.) is also used which represents the quantity of hi 
ed bo raise 1 3b. of water through 1'C. 

good account of these method * js given in Glazebroafe, A Dictionary of 
Physics, Vol. I, article on "Catorimetry". 



(») 



Methods based on Change of State or Latent Heat 
Calorimetry, 

(■1) Electrical Methods. 

In die following pages we shall discuss these methods one by 

der each of these we shall consider the various forms of 

mental arrangement that have been adopted. Solids and liquids 

will be considered first while gases will be taken up later in the 

iter. 

L METHOD OF MIXTURES 

3. Theory of the Method.- — Regnault* about the year 1840 made 
,i careful study of the Method of Mixtures, and by 'care and skill 

icd results of the highest accuracy. The principle of the m. 
is to impart the quantity of heat to be measured to a certain mass 
of water contained in a vessel of known thermal capacity and to 

tire the rise of temperature produced. Thus, if a substance of 
,72-,., specific heat s t and initial temperature 6 Xf be plunged into 

rams of water at temperature 0g* and if W be the thermal capa- 

of the calorimeter, the final temperature of the mixture, we 
1 1 i : . i. • , by equating the heat lost by the ice to the heat gained 

by the water and calorimeter, 

lis gives the specific heat of the substance. Various correc- 
r, necessary for heat is lost by the system by 
luction, convection and radiation. Thus lor 8 we must put 

' e /S& is* the correction. 

A. Radiation Correction, — In most experiments on calorir 
the calculation of this loss of heat due to radiation is important. 
ITic radiation correction may be accur- 
al Hy calculated with the help of 
ton's Law of Cooling (Chap. XI) & 
h states that for small differences 
of temperature the heat loss due to 
iation is proportional to the tempera- 
difference between the calorimeter 
i iul the surroundings. To illustrate its 
application kit AB (Fig. 1) denote the 
rved rise of temperature during an 
ent, RC the observed cool: 
i, I of it. We have to calculate 
rue rise in temperature*. Divide 
the abscissa into n equal intervals 5.',. 
:.i....St B by means of ordinate* P 1 M ii P 2 M 2 , . . .P„M S such that 

* Henri Victor Regnault (1810-1878), born at Aix-Ja-Chapclle, had to support 
himself while young, He joined the Ecole Poly technique in Paris and later on 
in [840 he was appointed Professor in that PoJyteehnique. He r.id many classic 
. on heat. 



AM.MW W« 



■t-f 



fl 



+ D 



Fig. 1. — T] lustration of Radiation 

CoiTU'/li: 'I. 



30 



CALORIMETRY 



CHAP, 



the small portions AI^ PJ* . .. P^P., may be treated as straight 
ines. Let us measure temperatures from th< 



ie temperature or the 
surroundings. If 8 X , 2 . . . denote the mean temperatures during these 
intervals, B t ', 0/ . , . the temperatures at the ends of these intervals 
represented by P^, P 2 M 2 . . „ then the temperature diminution due 
to radiation in the interval gr^ is kBtfh. If ${, 0*, . . . denote 
the temperatures at die ends of these intervals had there been no 
loss due to radiation, then 



0-"^ 



= 6, 



t "= #,' -- kB x U x 

a "= 0/ -f MjBtj, -|- ke.,bt 2 . 
+ Af^&i -!- 8 a/ 8 -f- .. .. dji„] 
+ £(area of the curve ABM„A) 



= V + k 



'j: 



e *//. 



(i) 






We can thus correct any temperature 0/ 
AP.M.A and k. 



if we determine the area 



An alternative method is to plot the upper curve from the lower 
curve by increasing the ordinate M t P x to MiP/, M,P. to M«P,' etc 
where M^P/ = ^ HP,'=V. etc. The highest ordinate "on the 
curve (wo. D£) gives the true rise of temperature in the experiment 
corrected for radiation. r 

To determine h we have to observe the rale of cooling at anv 
temperature. The curve BC {Fig, I) is obtained experimentally for 

this purpose. From this ~ is calculated for any mean value of $. 

is known.* 

Another method called the adiabatic method is to eliminate the 
teat bj continuously adjusting the temperature of the bath enclosing 
me calorimeter to be always equal to the temperature of the caTori" 
ureter itself.* 

5. Specific Heat of Solids. — For finding the specific heat of solids 
by this method the requisites are a calorimeter with an enclosure, a 

thermometer and a heater. For work at ordinary temperatures the 
calorimeter is made of thin copper, uickelplatcd and polished on the. 
outside., so as to reduce radiation losses, it is supported on pointed 
pieces of wood or by means of thread inside a larger double-walled 
vessel which has water maintained at a fixed temperature in the 

,.;, t,A« fl rt nple b v fc ro V; ?h m ! thod sometimes adopted is to add to the observed 
rise halt the coolwyr observed at the hi R hest temperature m a time equal to the 
duration .of the experiment. This is based on the assumption that the average 
excess of temperature of the calorimeter over the surroundings may be taken In 

«Uw i tt i T^ S > hL T cc the co ° ]in « durin ff the experiment is half th 
cooling at the final temperature. 

* tt F Ja other methods £cc Glazebrook, A Dkiiottary of Applied Physics, Vol. 1 



SPECIFIC HflAT OV LIQ1 '" ■ 



31 



D 



la space between the walls. The heater is a steap-jacket in 

,, i substance is heated by steam without becoming wet. An 

oil bath i i be used. The transference and radiation errors must 

i by suitable mechanical devices as in Rcgnault's classical 
in ents. 
For high temperatures the solid substance is heated in an electri 
furnace. While in his work at high temperatures employed a furnace 
h .. in •■ .■ platinum coil wound on its surface. The substance is sup- 
inside the furnace in a loop of platinum wire 
up! is allowed to drop into the calorimeter by a 
i i mechanical device. Change in temperature 
is measured by a resistance thermometer. For work 
m tow temperatures the substance is cooled down in 
a quartz vacuum-vessel surrounded by liquid air 
before being dropped into the calorimeter. 

Awbery and E/er Griffiths have determined the 

irk heat of solids and molten liquids as well as 

i latent heat by using an improved apparatus 

i iid on the method - of mixtures. This is discussed 

in Chap. V. 

The use of water as calorimetric liquid has 
eral drawbacks. Its range is small and specific 
large so that the rise of temperature is small ; 
lier there is considerable risk of some water being 
1 1 i,i by evaporation. For these reasons several workers 
have replaced it by a block of metal The copper 
block calorimeter devised by Nernst, Lindernann and 
Koref is exceedingly convenient for low temperatures. 
It consists of a " heavy copper block X (Pig. 2) 
.Hinted with Wood's metal to the inside of a Sewer 
flask D. It is essentially a calorimeter based on the 
method of mixtures in which copper replaces water 
as the standard substance. The heated substance is 
iped into die copper block through the glass-tube 
11 and the change in temperature of the latter is read 
on thermo-couples T, T, whose one end is inside the 
copper block K and the other end in the block G. 
The copper block on account of its good conductivity keeps the 
temperature uniform. Jaeger and his co-workers have employed this 
method to determine the specific heat of W, Vt, (X Rh, Ir, etc., to 
about, lGOn^C with a high degree of accuracy. 

6. Specific Heat of Liquids. — Specific heat of liquids which do 
not react chemically with water or any other substance or known 
specific heat may be obtained by direct mixture. For liquids which 
react in this way Rcgnault used a different form of apparatus. The 
liquid was not allowed to mix with water but was admitted when 
desired into a vessel immersed in water. The liquid was first heated 



•T- 



Ffc. 2. 

Copper Block 

Calorimeter, 



CAl.ORIMETRY 



[CHAP. 

The specific heat could be calculated 



and then forced into the vessel. 
as before.* 

Another cla*s ol experiments For measuring the specific heat, of 
liquids involves the expenditure af some mechanical energy and 
measurement of the consequent rise of temperature. To this class 
belong die classical experiments of Rowland for determining the 
ichamcal equivalent of heat. They will be described in detail in 
Chapter 11L 

2. METHOD OF COOLING 

7. This method, perfected by Dulong and Petit, is found to be 

most, convenient for liquids but unsuitable for solids owing to varia- 
tions of temperature within the latter. The method is based on the 
assumption that when a body cools in a given enclosure on account 
of radiation alone, die heat dQ emitted in the time dt is given bv 
die relation 

d(l=Affldt, . . . . ,(4) 

1 depends upon the area and the radiating power of the Em 
f{9) is an unknown function of $, the excess of the temperature of 
tile body over that of the surroundings. 

I! :' ; produce* a cooling; of the body through -d$, we have 

dQ = -msdd, 

iote the mass and die specific heat of the body 

ng these two expressions for dQ we get 
- made =3 Af(&) dt. 

Or f .. m * a dO 

Or t __ ms f* 1 ii8 _ 

" * )&*Mr ' ' - ' - I 5 ) 
e * is the time the body takes in cooling from X D to a 2. 
Similarly for another substance to cool through the same" interval 

Ml LcMl'J J _lcl <X Lll] C 

' A' )B s f(d)' ' ( 6 ) 

* i- ~: r i e * Che sutfa f e area and the radiating power- of the 
two bodies be rbx: same we have from (3) and (6) 

ms m's' 

t =- r- ■ ■ - (7) 

If masses m, m> of two liquids be contained successively in a- 

calorimeter o| thermal capacity W and the calorimeter Upended 

* For details see Preston, Theory of Hmt 



u 



MFTUOD OF MILTING ICE 



, 



inside a vessel kept at 0°C by immersion in melting; ice and then 
.a lions of the rate of cooling taken, we have 



W_ ■ 
t 



*o 



(S) 



II' one liquid is water (s — 1) , the specific heat of the other is thus 
determined from a knowledge of t, t', m t m'. 

The method is sometimes employed for determining the specific 
li it of liquids but is not capable of any great accuracy and is mainly 
of historical interest. 

3. METHODS BASED ON CHANGE OF STATE 

8. These methods may be subdivided Into two : namely the 
method of melting ice and the method of condensation of steam. 
These methods were of real advantage in the last century when 
accurate measurements of temperature were impossible, but with the 
recent development of accurate thermometers and electrical heaters 
they are now less in use, chiefly on account of their inherent defects. 
The second method is, however, very convenient for determining the 
specific heat of gases at constant volume and hence retains its 
importance. 

9. Method of Melting Ice.— In this method the heat given out 
by a certain substance in cooling is imparted to ice and measured by 
the amount of ice thereby melted. Thus ir M .grams of a substance 
of speciQc heat a and initial temperature & are able to melt m grams 
of ice when placed in contact with the latter, the specific heat is 
given by the relation 

MsB — m.L, . . . . 

where L is the latent heat ol; fusion. The earliest forms of the appa- 
ratus as devised by Black, and by Lavoisier and Laplace were liable 
to cause considerable error. An improved form of the calorimeter 
was later devised by Eunsen* which will now be described. 

10. Bunsen's Ice Calorimeter. — In this calorimeter, the water 
produced by the melting of ice is not drained off but is allowed to 
remain mixed with ice and the resulting change in volume is observed. 
The calorimeter is illustrated in Fig :<. p. 34. The test tube A is fused 
into the cylindrical gla^s bulb B which is provided with the glass 
stem C. B is nearly filled with boiled air-free water and the remain- 
ing spare and the stem is filled with mercury. The stem terminates 

an iron collar D containing mercury into which a graduated 
tillary tube E is pushed so that mercury stands at a certain 
• n in E. 

In conductio , an experiment a stream of alcohol, cooled by a 
ezing mixture, is first passed through the test-tube A until a cap 

♦Robert Wffliebxi Bi 111-1899), born at Gottingea, studied at Gottingen, 

Paris, Bcriin and Vienna. He r of Chemistry at Bresfau and 

berg. His important researches arc on spectrum analysis Bunsen cell, 
'. i;- •:•••- gas tntni :r and ice calorimeter. 

3 



34 



CALORIMKTRY 



[CHAP. 



of ice F is formed round it in B. The whole instrument is then 

kept immersed in pure ice at 
0°C for several days till all the 
water in B is frozen. It is then 
ready, for use. 

To calibrate the scale on E, 
let a mass m of water at a tem- 
perature 0°C he poured into the 
test-tube. Some ice in B 
melts and the resulting contrac- 
tion of mercury, say n divisions 
on E, is observed. Then if a 
recession of mercury by 1 divi- 
sion corresponds to q calories of 
heat 

rn$ = nq ; or q = mQ/n .,■ (10) 

Next the substance un4er 

i vivc:, ligation, previously heated 
to a temperature &', is drooped 
into some water at 0° contained 
in the test-tube A. Then if M, 
Fig. 3^-Bui Calorimeter. s d enote the mass and the spe- 

tanoe respectively, and v? the observed reo 
oJ mercury thread in E, we have 

Mse' = n'q, 




or 



m q " 



Mnff 



(11) 



ill ilic heat of the substance. The specific heat, 

of rare metals which can be had in small quantities can be readily 

I by this method. The apparatus is, however, not capable of 

-rear accuracy. A fundamental objection to the use of the ice 

calorimeter rests on the fact that a given specimen of water can 

i: into ice of different densities, 

11. Joly's Steam Calorimeter,— In the steam calorimeter devised 
by Prof, Jolv in 1886 the heat, necessary to raise die temperature 
of a substance from the ordinary tempera Lure to the temperature of 
steam is measured by the amount of vapour condensed into water at 
the same temperature. It consists of a thin metal enclosure A 
(Fig. 4) , double-walled and covered with cloth, which is placed 

lath a sensitive balance. One pan of the balance is removed and 
from this end of the beam hangs freely a wire w supporting a platinum 
pan inside the enclosure- The substance whose specific heat is required 
is placed on this pan and weights added on the other pan till balance 
is attained. The temperature of the enclosure is observed by means 
of a thermometer inserted into the chamber, and in the meantime 



tL] 



jqly's steam calorimeter 



35 



steam is prepared in the boiler. It is then admitted suddenly into 
the chamber through the wide opening O at the top and can escape 




Fig. 4. — Job's Steam Calorimeter, 
through the narrow exit-tube t at the bottom. Steam condenses on 
the substance and the pan, and weights are added on the other pan 
to maintain the equilibrium. When the pan ceases to increase in 
weight the readings are noted and the temperature of the steam read 
on a thermometer. During the final weighing the steam is all 
to enter die chamber through a narrow escape-tube so as not to disturb 
the pan. The weight becomes practically constant in four or five 
minutes though a very slow increase of about 1 milligrams per hour 
may be observed due to radiation. The difference between the two 
weighings gives the weight of steam condensed. 

If W is the weight, of the substance, w the increase in weight 
of the pan, $ t the initial temperature of the enclosure, $ s the tempera- 
ture of steam, k the thermal capacity of the pan, L the latent heat 
of steam, s the required specific heat, then 

k is determined from a preliminary experiment without any substance 
on the pan and thus the specific heat of the substance is found. 

For great accuracy •various precautions and corrections are neces- 
sary. Steam condenses on the suspending wire where it leaves the 
chamber and then surface tension renders accurate weighing difficult. 
A thin spiral of platinum wire in which a current Hows, usually 
surrounds the suspending wire just above the opening, and is made 
to glow so that the heat developed is just sufficient to prevent con- 
densation, A rapid introduction of steam is necessary in the early 

s, for .steam also condenses on the pan due to radiation to the cold 
air and the chamber, thus causing error. This is, of course, partially 
balanced bv radiation from the steam to the substance later. 




3G 



CAL0RIMETRY 



[< a ip. 



Further w does noi accurate!}' represent the weight of si earn con- 
densed since the first weight is taken in air at 0-^Q and the second 
in steam at Ss°C. All the weighings must be reduced to vacuum 
and then the increase in weight calculated. Specific heat of raw 
substances ran be found by this method since small quantities ui 
the substance are needed but a sensitive balance is indispen; 
The specific heat of liquids and powders can be found by enclosing 
them in glass or metal spheres whose thermal capacity is taken into 
account. Cases can also be similarly enclosed, but then the modified 
form of the apparatus — the differential calorimeter — is us< 

12* The differential Steam Calorimeter, — -In this form invented 
by Prof. Joly in 1889, both the balance pans are made exactly 

similar and of equal thermal capacity 
and hang in the same steam-chamber 
(Fig. 5). The substance to be tested 
is placed on one pan and the ex- 
cess of steam condensing on this pan 
over that on the other pan is entirely 
due to the substance. Thermal can. i 
of the pans, radiation from them and all 
other sources of error common to them 
are eliminated, the substance bearing 
only its own share of the error. The 
chief use of this apparatus, hoy 
consists in the determination of the spe- 
cific heat of gases at constant volume. 
The pans are then replaced by two equal 
hollow spheres of copper furnished with 
"catch-waters" (shown in the figure), 
One sphere is filled with the dried 

S mental gas at any desired 
sure while the other is empty. These 
spheres are counterpoised by adding 
necessary weights m which represent 
the mass of the contained gas. St* 
is admitted and condenses on the pans. 
A larger amount of steam condenses on 
the sphere containing the gas, the 
excess, say w t giving" the amount of 
steam required by the gas. Now the 
specific heat, at constant volume c„ may 
be calculated from the equation 

'fy— e t )=wLt (13) 

e $& 3 are the final and initial temperatures of the chamber. 
Prof. Joly used copper spheres of diameter 6.7 cm. and weighing 

pa. and employed gases at different pressures. Com tio 
applied for the following : — 

1. The expansion of the sphere due to increased temperature 
the consequent, work done by t,lte gas in expanding to this volume. 




: - itiHl St. 'run 

Colorimeter. 



ll.j 



iiUiCTRl'CA I . METHODS 



o7 



2. The expansion of the sphere due to the increased pressure 
mi the L;as at the higher temperature. 

le thermal effect of this stretching of the material of which 
the sphere is made. 

4. The increased buoyancy of. the sphere due to its increased 
tie as die higher temperature. 

rhe unequal thermal capacities of the spheres. 
6. The reduction of the weight of water condensed to its weight 
acu 

Dewar has devised calorimeters based on an analogous principle in 

lie employed a liquefied gas as the calorimetric substance. Tiie 

t. to be measured is applied to the liquefied gas whereby the liquid 

i vaporates absorbing its latent heat and the volume F of gas thus 

duced is measured. The heat communicated to the liquid is then 

en by V pL where p is the density of trie vapour and /. the latent 

at of vaporization of the substance. Using liquid oxygen and liquid 

d ogen the apparatus can be adopted for very low temperatures. 

1 1. the case of hydrogen 1 c.e. of vapour at N.T.P. corresponds to a 

small quantity of heat (about 1/100 calorie). This method has 

id for measuring the specific heat down to very low tempe- 

The experimental substance (solid or liquid) is first kept in 

itant temperature bath (say 0°C) and then dropped into the 

containing liquid oxygen or liquid hydrogen. 

4, ELECTRICAL METHODS 

13. The electrical method was first employed by Joule in his 
npts to determine the mechanical equivalent of heat. The 

trical methods at present available may be subdivided into two : — 
(1) Method based on the observation of rise of tempera: 
:'') Method employing the steady-flow electric calorimeter. 
We shall first consider the application of these methods to liquids 
mse historically the method was first applied to them. 

14. Methods Based on the Rise of Temperature. — Following Joule 

this method was adopted by many workers the chief among them 

being Griffiths, Schuster and Gannon, W. R. Bousfield and W. E, 

BouSfield. They employed Bliss method for determining tile mecha- 

lical equivalent of heat and found that it was capable of the highest 

■ uracy. The same arrangements may be employed for finding the 

he heat of liquids. 

The principle of the method is to generate heat by passing a 

renl through a conducting wire. If i is the current through the 

wire of resistance R and E the potential difference across its ends, the 

spent in a time t seconds is Eit ergs, provided K and i are 

ed in electromagnetic units. If this raises the temperature 

ol M grams of a substance by M°, the specific heat s of the substance 

is given by the relation 

EU—JMsM, (14) 

ere / is the mechanical equivalent of heat (see Chap, J if). If E 



38 



CALORIMK'fltV 



[chap. 



is expressed in volts and i in amperes the energy spent is give 
| anles (1 Joule = 10 7 ergs.). 

Any two of the quantities E, i and R may be measured, Lhus 
giving three methods. Griffiths, in his determination of the sp 
neat of water, chose to measure E and R which is rather difficult 
for R must be measured during the heating experiment, GiiffiLhs' 
work is important since it first established the fact that the electrical 
method can accurately give the value of J in absolute units, Schuster 
and Gannon measured E and L 

Jaeger and Steinwehr have applied this method Lo determine the 
mechanical equivalent of heat and hence also the specific heat at 
different temperatures. They employed a large mass of water (50 
kg.) and consequently the thermal capacity, of the vessel was only 
about 1% of that of the contained water. A section of their apparatus 
is shown in Fig. 6. AA is the cylindrical copper calorimeter lying 
on its side and properly insulated from the surrounding constant- 
temperature bath B. On the 
upper side at O there is a hole 
for the introduction of the 
heating coil H t the resistance 
thermometer and the shaft, t 
which drives the stirrer SS. A 
_nt. of about 10 amperes 
was allowed to flow for six' 
minutes through the const a n- 
tan heater H of S ohms resis 
tance and the rise in tempe- 
rature was about 1.4 °C In 
v experiments an accuracy 
in 10,000 was aimed at 
and hence the results are very 
reliable, 

IS. The Method of 
Steady-flow Electric Calori- 
meter,— Great accuracy was 
ained by Callendar and 
Barnes by using the steady- 
How electric calorimeter shown in Fig, 7. A steady current of the 
experimental liquid Sowing through the narrow glass-tube /, about 
2 mm, in diameter, is heated by an electric current flowing through 
the central conductor of platinum. The steady difference of tempera- 
ture SB between the inflowing and outflowing water is measured by 
a pair of platinum thermometers Ft, Pt at each end connected differ- 
entially in the opposite arms of a bridge of Callendar and Griffiths' 
type, 'The bulb of each thermometer is surrounded by a thick copper 
tube of negligible resistance attached to the central conductor. This 
on account of its good conductivity keeps the whole bulb at the 
temperature of the adjacent water, and due to its Ion- resistance 
prevents the generation of any appreciable amount of heat by the 




Fig, 6. — A Section of jaeger and 
Steinwehr' 5 Calorimeter. 




STEADY-FiLaW ELECTRIC CALORIMETER OV 

current near the thermometer. The leads L, L and P., P are attached 

to this tube of copper, die former for introducing the heating current 

and the latter for measuring the potential difference across ihe central 

uctor in terms of a standard cell by means of an accurately 




Fig. 7.— Steady-flow Electric Calorimeter. 

calibrated potentiometer. The potentiometer also serves to measure 
the healing current i by measuring the potential difference across 
a standard resistance included in the same circuit. In order to 
diminish the external loss of heat the flow tube is enclosed in a herme- 
tically sealed glass vacuum jacket surrounded by a constant tempe- 
rature bath. Neglecting small corrections the general equation is 

Eit=JMs (ds-dj+jkt, .... (15) 

employing the same notation as before, where h denotes the heat 
loss per second on account of radiation, and & if &<, the temperatures 
of inflowing and outflowing water. The time of flow t in these experi- 
ments was about 20 minutes and was recorded automatically on an 
electric chronograph reading to 0.0 1 sec. The mass of water M was 
measured by collecting the outflowing water and was about 500 gm. 
The difference in temperature Q*—9t was from 8° to 10 D C and was 
accurately read to -001 n C. The heat loss h was very small and 
regular, and was determined and eliminated by suitably adjusting the 
i trie current so as to secure the same rise of temperature for 
different rates of flow of the liquid. Thus for two rates of flow we 
have 

E^t=JM v t{8 t -d x ) -f- Jht, 

•' J iM x -M t ) [Ot-W 

Since the temperatures at every point of the apparatus are the same 

in both experiments the heat loss h must also be the same. The 

i ecific heat s thus determined is the average specific heat for the 

interval W and may be taken as the specific heat at the middle of 
the interval. 

The great advantage of this steady-flow electric method is that 
no correction is necessary for die thermal capacity of the calorimeter 



40 



CALORIMri'RY 



[CHAP. 



since there is no change of temperature in an) part of the instrument 
Care must, however, be taken to secure perfeci steadiness, as it is 
practically impossible to correct for unsteady conditions. Further, 
since all condition?, are Steady, the observations cart he taken with 
tiie highest degree of accuracy. There is no question of thermo- 
metric lag. It is essential, however, that the current of water be 
thoroughly mixed otherwise temperature over a cross-section of the 
tube will' not be uniform. This is secured by having! the central 
Wictor in the form of a spiral instead of a straight wire, 
Callcndar and Barnes used this method to find the specific heat 
of water at various temperatures. Their results are discussed in the 
next, section- Callendar found the specific heat of mercury by this 
method. The central conducting wire was dispensed with, die flowing 
mercury itself serving as the conductor. Griffiths employed this 
method to determine the specific heat of aniline over the range i 
to 50°C. 

16. Specific Heat of Water. — In ordinary calorimtftric experi- 
ments the specific heat of water is assumed constant, at all tempera- 
tures and equal to unity. Accurate investigations of the last section, 
show that it varies with temperature. The first accurate experiments 
in this connection were those of Rowland in connection with his deter- 
mination of the mi' ical equiv: leni " : heat (Chap. 111). He argued 
e specific heat of water at all temperatures were constant 
this mechanical equivalent must come out a constant quantity even' 
used water at different temperatures. The variation in the value 

irk heat. 
?■ at different le\ wres. 



mp. 


Callendar 


Jaeger and 


Osborne 


Specific heat 




arnes 


Stemi 


Stimson & 


5n int. Joules 
(O, S, & G.) 





1.0093 


(1.005) 


1 .0076 


1 4.2169 


.1 


1.00-f7 


1.0029 


1.0O39 


1.2014 


10 


1.0019 


1.0013 


1.0015 


4.1914 


15 


1 .0000 


1.0000 


oooo 


11850 


20 


0.9988 


0.9990 


9991 


4.1811 


25 


0.9980 


0.9963 


0.9fl 


4.1788 ' 


30 


0.997ti 


0.9979 


9982 


4.1777 


35 


0.9973 


0.9978 


0.9982 


4.17. 


40 


0.9973 


0.99R1 


0.9983 


4.1778 


45 


0.9975 


0.9987 


0.9985 


4.1787 


50 


0.9978 


0,9996 


0.9988 


4.1799 


60 


0.9987 




0.9997 


4.1836 


70 


1.0000 




1.0009 


4.1888 


80 


LOO 17 




1.0025 


4.1956 


90 


1.0036 




1.0046 


4.2043 


100 


L0057 




1.0072 


4.2152 



iu 



NERNST VACUUM CALORIMETER 



41 



The oilier accurate experiments on the subject are those 
sndar and Barnes (sec. 15) and of Jaeger and . Steinwehr (sec- 14). 
Both of them determined accurately the specific heat of water at 
us temperatures, Their values are given in Table 1 together 
with the value;, obtained recently by Osborne, Stimson and Ginnings 
at; the National Bureau of Standards, Washington. In column 5 the 
specific heat is expressed in international Joules* per gram per °C. 




40 s ST 

TSUPERAI b*RE K 



II-!..' 



Ft- 8.— Specific Heat Curve far Water. 

he results of all these three investigations arc plotted in Fig. 8. 
It will .he seen thai the values obtained by Callendar and Barnes lie 
somewhat wide of the others and appear to be less reliable chiefly on 
account of the uncertainty in the values of the electrical units 
employed. From these curves it is evident that water has a minimum 
ific heat at about 34 n C. It is on account of this variation that on 
IB the calorie was defined with respect to 15 C C. 

Specific Heat of Solids 

17. Rise of Temperature Method.— The electrical method was 
first applied to solids by Gaede in 1902. E. H. Griffiths and E. 
Griffiths determined the specific heat of many metals over the range 
- 160° to | 100°C. The substance was used in tire form of a calori- 
meter and was first cooled below the desired temperature. Electrical 
energy was utilised in heating the calorimeter and the temperature 
led by a resistance: thermometer. The calorimeter was enclosed 
in a constant temperature bath whose temperature was kept constant 
to 1 /100th of a degree. Correction was applied for the heat lost by 
radiation, 

»yl8. Nernst Vaeuam Calorimeter.— A different form of the appara- 
tus, known as the vacuum calorimeter, was used by Nernst and 
Lindemann for measuring die specific heat at very, low temperatures. 
This differed from Gaede's form essentially in having the calorimeter 
suspended in vacuum. The results achieved with its aid are of great 
theoretical importance and hence their apparatus will be considered 
* 1 /Int. Jodie = 1.00041 X 10 7 ergs. 




CALORJMETRY 



CHA '. 



in some detail, T?or good conducting solids such as metals the 
calorimeter shown in Fig, 9(a) was used. The substance whose 
specific heat is to be determined is shaped into a cylinder G, having 
a cylindrical hole drilled almost through its entire length, and a closely 
fitting plug P made for it from the same material. The substance 
here acts as its own calorimeter. The plug is wound over with a 






p 



c 




fa) ( b) (C5 

Fig. 9.— Nernst Vacuum Calorimeter, 

spiral wire of purest platinum (shown dotted in the figure) which is 
insulated from it by means of thin paraffined paper, and finally liquid 
paraffin is poured over it. The upper part of the plug is somewhat 
thicker than the lower part, thus a good thermal contact is obtained. 
The calorimeter K thus constructed Is suspended inside a pear-shaped 
glass bulb [shown at (b)] which can be filled with any gas or evacuated. 
The whole can be surrounded by suitable low temperature batlis such 
as liquid air or liquid hydrogen. The platinum spiral, which serves 
both as electric heater and resistance thermometer, is connected in 
scries with the battery B, resistance r and a precision ammeter A, 
the voltmeter V indicating the potential difference across the spiral. 
In order to bring the calorimeter to the desired tempeiature of 
experiment., hydrogen which is a good conductor of heat was first 
admitted into the pear-shaped vessel and the latter surrounded In a 
suitable bath. Next the vessel was completely evacuated so that the 
heat losses from conduction and radiation were almost entirely eli- 
minated. Tn addition it was surrounded by liquid air or liquid 
hydrogen. 




it. 



RESULTS or EARLY EXPERIMENTS 



experiment a current was allowed to flow through 
xnds and the voltage across it was adjusted to be 



denote 



To carry out an 
the heater for I seconds and the voltage 
constant by varying the resistance r. If Rf, Ri and i fs 
the final and initial values of the resistance of the heater and the 

enl through it respectively and E the constant potential difference. 

Thus an observation of it, i t and E gives R f and R, » and from a 
previous determination of the resistance oE the platinum spiral at 
various temperatures the rise in lemperature 80 can be found. The 
energy supplied electrically is Eit where i is the average value of the 
current. Now if M is the mass of the substance forming the calori- 
meter, s its specific heal, we have, 

Eit = JMsM -f h. . . . . (15) 

This gives the specific beat at a single temperature since BO is usually 
1° or 2°. The heat capacity of the paper and paraffin can be found 
and eliminated by taking different amounts of the substance and at 
the same time arranging that the temperature rise is the same The 
heat loss h is very small and is determined and accounted for I - 
observing the rare of cooling before and alter the experiment. 

For non-conducting solids' the calorimeter shown in Fig. 9 (c) was 
employed. The heating w T ire was wound over a cylindrical silver 
vessel D and the whole covered with silver foil to diminish heat loss. 
This foil was soldered at the bottom of the cylinder as indicated. The 
solid whose specific heat is required was placed inside the silver 
cylinder and the latter closed with the lid. The silver on account of 
its high conductivity keeps the temperature, uniform and this is 
further secured by filling the cylinder with air through the tube in 
the lid. The tube is then closed with a drop of solder so thai it 
mav be gas-tight. It is absolutely necessary that air should be 
sent inside the vessel to facilitate equalisation of tern pern lure 
throughout, the experimental substance. Liquids and gases can be 
similarly admitted into the cylinder and their specific heat determined. 

19. Results of Early Experiments.— In 1819 Dulong* and Petit 
from their investigations concluded that the product of atomic, weight 
and specific heat 1 xf/as constant for many substances, or in other words, 
ims of all substances have the same capacity for heat. Regnault 
from his own researches found that for ordinary substances the mean 
value of the constant was 6.38 with extremes of 6.76 and 5.7. A 
more accurate value of the constant can be obtained from the kinetic 
theory (Chap. III). The atomic heat at constant volume is shjwn 
there' to be equal to 31? = 5.955. According to Richarz, the value 
::T the ratio Cp/c e for many substances lies between 1.01 and 1.04, 
hence the atomic heat at constant pressure, the quantity commonly 
determined should lie between 6,01 and 6.19. This law T is of great 

* Pierre Louis Dulongr (1785-1838), a distinguished French scientist who lost 
an eye and a finger owing to the explosion of some nitrogen chloride which he 
discovered. 






CAI.ORIMETRV 



CHAP. 



aiming atomic weights. Tn illustration of the law table 4 
- is adde . 

Ta ble 2 — Illustration of Dnlong and Peti 



Element 



am 
M; '_ 

AJuminiitiii 
Iron 

led 
Copper 
Zinc 
Silver 

lium 

Antimony 

LI 



Atomic Weight 


Mean specific 


. 


(1) 


IlC Hi 


heal 






: (2) 


23*00 


0.307 


7.06 


L32 


0.247 


li.nii 


27.1 


0.2175 


5,83 


55 


0.110 


6.14 


.68 


0,1092 


6.41 


63.57 


0.0930 




6&,37 


0.0939 




107.0 


0.0559 


6.03 


11. 


0.05 


6.26 


118.7 


0.0556 




121.8 


0.05 


6,10 


[95 


0.0318 


6.21 


197.2 


0.031 


8.10 




0.0310 




209 


0.0299 





ies= 6.24 

,n mn enunciate aw concerning molecular 

heat and 

onstant The 
- ; ll "' : " ' mounds to another In 

illustration of the tbkf 3 is added. 

—Moteo at of Oxides. 



Compound 



i tfic heat 



Cr s O a 
Bi a O, 



Moleculai . Secular 

-dit heat 

(2) 



0.1700 
0.1796 
0.1277 

0.0901 

0.0605 



159.8 

192.0 
197,8 

287.fi 



27.2 

27. J 

. 

25.9 



Mean value = 26.8 

• ! :en f rom Vol S n 193 

' lake!1 : Experimental ! ' g p , 200, 



■". 



TWO St'I CI! [CI 






Limann's law can be considered as a particular case oE die 
following law : 1 i ular heat of a compound may be considered 

is the the atomic heats ol Its constituents. Thus if a com- 

pound has the composition A a "&i,G c T)j its molecular heat C t is given 
n the relation 

C p =aC,, , D . . . . (17) 

B, C, D stand for the different types of atoms composing 
the compound and C pjLt C p1i , etc., their atomic heats given by 
mlong and Petit's law. The law is ol much use in evaluating the 
nolecular he. [lain substances, 

20. Variation of Specific Heat with Temperature. — The Specific heal 

determined by the foregoing methods is not found to be a constant 

juantity. For solids and liquids the effect of pressure on specific 

i rather small. The effect of temperature is however very 

d. Increase of temperature invariably -increases the specific 

feat, while the decrease of temperature lowers it. In fact the atomic 

lea solid almost vanishes at the absolute zero and gradually 

increases with rise of temperature reaching asymptotically the Dulong 

bid Petit's value of 3JS at a sufficiently "high temperature which is 

Iiflerent for different substances. This variation for silver is illus- 

d in Table 4. 

—A torn ic heat of silver at different temperatures. 



Temp, in c Iv 



1.35 

S 
10 

20 

36.16 




0. i:'V 

0.00509 
0.0475 
0.399 
1.694 



Temp, in °K. Atomic heat 



55.88 

74.56 

103.1 I 

205,3 



3.186 

:.u;'-| 

5.373 
5.605 



The asymptotic value oF $R can be accounted 1 for by the Kinetic 
Theory (Chap. Ill) . The variation with temperature has been 
successfully explained by the quantum theory of specific heats and 
in particular by the Debye's theory of specific heat which is how 
beyond the scope of this book. 

21. Two Specific Heats of a Gas. — The specific heat of a gas. 
as of solids and liquids,, may be defined as the ratio of die heat 
bed to the rise in temperature, taking a unil mass of the gas. 
A little consideration will show thai Emition requires to be 

1 'ne a quantity of gas to be suddenly compressed. 

The ' ii : oJ the gas will be found to rise, though no heat has 

been added. The rati •' added/increase in tempera' v. 

ic heat, vanishes. Again let this com] air expand 

; ]]'.. ; :\ roolrng v. ;ikc place. Tins is just prevented bv 

applying some heat to the gas. In this en.se the ratio, heat ad 



415 



CALORIMETRY 



CHAP. 



EXPERIMENTS OF GAY-LUSSAC & JOULE 



47 



change in temperature becomes infinite. Thus we see that the original 
definition gives an infinite range o£ values for the specific m 
Hence external conditions are of paramount importance in deter- 
mining- the specific heat of gases. It has become customary to 
ak of two specific heats o£ a gas : the specific heat at constant 
volume denoted by C, and the specific heai at constant pressure 
denoted by tip. In the former process the gas is maintained at 
constant volume so that the whole heat applied g es to increase the 
internal energy of the gas. In the latter case die gas is allowed to 
expand against a constant pressure and in so doing it does external 
work. This work is obtained by using up part of the heat energy 
applied ^ to the gas. Hence the specific heat at constant pressure is 
necessarily greater than the specific heat at constant volume by an 
amount which is simply equal to the thermal equivalent of' the/ 
work done by the gas. 

Let us assume that the gas is perfect, i.e., its molecules exert no 

influence on one another. This is approximately true for the permanent 

gases as Joule's experiment (Sec. 22) shows. Hence in this expansion 

no internal work -against molecular attractions is done by the gas and 

the excess of heat supplied in the second case is simply the thermal 

equivalent of the external work. Consider the gas enclosed in a 

vessel of any shape and suppose the walls of the vessel can expand 

wards. Lei &A denote an element of area of the walls and Bx the 

nice traversed hy, it measured along its outward drawn normal. 

the work done by the gas in this expansion against a cons- 

' pressure :.j^; and for die expansion of the enth 

the work is equal to % p&AM = pl&AJIx — p&V where 

lie volume of the gas. Suppose a gramme- 

ping a volume V t at temperature T°K and 

pressure to a volume V-, at temperature (T -|- 1) Cl K the 

og constant Let Mc p = C p , Mc p = C, where M h 

the molecular weight of die gas. C* and C, may be called trram- 



22. 



Gay-Lussac first car- 
internal 



■nol.- molar specific heats, 

expanding from V x to l-% is 



The work done by the gas in 



P(V S - V t ) ^ R [ (T +]}-T]=R J . . (18) 
y the gas laws. The thermal equivalent* of this work is T(C* — C ) 



■ . . (19) 
and C. measured in 



Hence 

JfC^-C,) =Rf, 

Or, if either R is expressed in calories, or C* 

. : ■•■■ we get 

C P -C S =R. 
This relation was first deduced by R. Mayer. 

if] oS^mS^. FirHt Taw ° f T^^anfe which will be diseased 
tThia is true for a perfect sas only. For real gases it cau be shown that 

Cp ~ CB = T \dT) L ,\ arJpWhere the differentials 



I be evaluated from the- 



Experiments of Gay-Lussac;!: and Joule. 
nc«| out experiments to determine whether a gas does any internal 

in expanding. He allowed gas contained in a vessel at a high 

o expand into an evacuated vessel and observed the fall of 

[en perature in one and the rise of temperature in the other vessel. 

is called free expansion or Joule expansion. In this the gas as 

whole does no external work while its volume increases, and the 

work done will be that against molecular attractions which is 

nal work. Somewhat later Joule employed a similar 

iratus but he immersed the vessel in calorimeters. His apparatus 





actual equation o£ state (Chapter IV) for the gas. 



Joule's experiment with Joulc:^ experiment with 

one calorimeter, two calorimeters. 

Fig. 10 Fig. 11 

indicated in Figs. 10 and 11. The two vessels A and B corn- 

" i a ted with each other through a tube furnished with a stop-cock 

A was filled with dry air at 22 atmospheres while B was exhausted. 

prat the whole apparatus was placed in a single calorimeter (Fig. 10) 

md the stop-cock C opened. No change in temperature of die water 

fas observed showing that no internal work against molecular attrnc- 

done I the gas in expanding. To investigate the point 

iriher, the parts A, B/C were placed in separate vessels (Fig. II;, 

attaining water whose temperature could he read by sensitive 

lermometers. On opening the stop-cock C the air expanded into B 

id the temperature of the vessel surrounding A fell while the tern- 

>eratures of B and C rose. It was found that the heat lost by A 

/as exactly equal to the sum of the heats gained by B and C, 

jus the total change in the internal energy of the gas during expansion 

zero. *.e\,\l— J — where U is the internal energy and P 

ic volume. This is called Joule's law or Mayer's hypothesis. This 
hows clearly that no internal work is done by a gas in expanding. 
[oule's law holds only for the perfect gas to winch, the permanent 
jases of Nature like helium, hydrogen, etc. approximate. For further 

J Louis Joseph Gay-Lussac (1778-1850) was a distinguished French scienti 
fho investigated on the expansion of gases. He was interested in aviation and 
1804 made a balloon ascent for the purpose of making experiments,. He was 
a peer of France. 



CALORiM ' 



[CHAP, 



iSUKEMENT OP C* 



49 






discussion see Chap, \ I, sec. lf> and Chap. X.. sec 3. As a n 
of fact, a slight fall in temperature should be observable in Ji 
experiment with one calorimeter but on account of the I. 
capacity of the calorimeter it escaped detection, 

23. Adiabatic Transformations. — When the pressure and volume 
of a substance change but no heat is allowed to enter or leave it, 
the transformations are said to be adiabatic (a = not, dia = thro 
bates = heat, i.e» 3 heat not passing through) • In an isd hermal i h 
thi temperature is kept constam by adding heat to or taking it away 
from the substance. Consider an amount o[ heat 8Q applied to a 
perfect gas. This is spent in raising the temperature of the gas and 
in doing external work. H we consider i jramnn olecule of the 
gas, the former is equal to C r d'f and the latter equal ro pdV/J, both 
in calories. Hei,. 

MZ = C,dT + pdV/J, 

or, if Sf) and C., are measured in ergs, 

= C,dT + pdF. ... 

This equation combined with pV = RTyn\l give the solution oi 
all problems on perfect gases,* In an adiabatic transformation 

80 = 0. Therei 

C B dT- r pdV = i 

in order to find a relation between p and V we must eliminate 
M (21) I jas equation pV = RT. DifTerentii 

Vdp = RdT i 

dT from (22) in (21) , we get 

and re pi ii by C p -C„. 

C,Vdp + C p pdV = 0, 
•'ting C p jC 9 by y we obtain 






dV 



i>« 



which on integration yields 

log p + y log V = constant, 

or — constant, 

is the adiabatic relation between p and V for a perfeel gas. 
find the adiabatic relation between 7" and V or bet reen /'and 
p we must respectively eliminate p or P betwei n ind 

the gas equal 

j-pv-i__ constant (24) 

■ — constat 1 1 i 

In ■"." bRT, and CV-C I substitttl 

equation oi state mid the true value oi C,-C„ We 
ion between />, F and 7 for real i^aes. 



, ._Drv air enclosed at 25 e C and ai atmospheric pressure 
,, 1 ,.] to half its volume. Find (a) the resulting 
the resulting pressure. Assume y = 1.40. 
!4) T 1 = T(7/7 1 )-- 1 -(273+25) (2)^ ~393«K. 

: 'I \V x )v--pV^ ut =2-64 atm. 

24. Experimental Methods*.— Let us now consider the expej 

ods oi Ending the specific heat of gases. Since for 

ises C„ -C B — R, a knowledge of one of the specific heats 
other. Again, if we dejpmine y, i.e., C I ie above 

and C B . Hence the experimental methods 
ivided into three classes : 

(a) The measurement oh (.',, 

(b) The measurement oE C,, 

(c) The determination of y. 

: MEASUREMENT OF C p 

25. The specific heat at constant pressure has been found either 
by the Method o£ Mixtures or by the Constant-Flo thod. The 
principles of these methods have already been explained. 

26. Method of Mixtures— Regnault's Apparatus. The method was 
first applied to gases by Lavoisier and Laplace. Improvements were 



GE2T 




I •!;■ imauh's Apparatus for CV. 

'■ The reader will find a very good account of these meQwxLg in Partingti n 
Heai oj Ga 






50 



CAUHUMi 



CHAP. 



made later by Delatoche and Berard, Haycroft and Regnautt. 

Regnault with his groat, experimental skill obtained results of high 

acy. His apparatus is indicated in Fiu;. 12. Pure dry gas was 

compressed in Lhe reservoir A which was immersed in a thermostat. 

Mi. reservoir was provided with a manometer (not shown). Gas 

could be allowed to flow through the stop-cock V at a uniform rate. 

This was effected by continuously adji Ele stop-cock V (shown 

■ rely) so that the pressure indicated jry the manomel r R was 

constant, The gas then flowed through a spiral S [n : in a 

hoi nil-bath and then into the calorimeter C fir:: ; into 

lhe air. The gas acquired the temperature T oj rath and raised 

the temperature of the calorimeter, say, from (^ to i),. 

Tf m is the mass of the gas that flows into the calorimeter, r^ its 
specific heat, is given by 

(26) 



mo (r A t*)-»(V- 



w is the thermal capacity of the calorimeter and it* contents. 

I 1 1 .: : mass m of the gas was determined by Regnault as follows; — 

f€>r any pressure p he assumed that the weig lined 

in the reservoir at temperature was given bv the relation 

W(l + a&) =Ap+.r: C 

B t C were determined, from leliminary experi- 

rresponding to the observed 
dent is given 
\p> -f- Bp 

of the gas iras found.* 

i>i gain <■'• 
and radiation which have to be found out 
.tare of the calorimeter before and 
the experiment, and taking the avea 

27. Other Experiments, The experiments of Wiedemann v 

similar to those of Regnault. Lussana devised a high- 

ire apparatus in which the same amount of gas enclosed at 

a high pressure can he repeatedly healed and passed through the 

The principle employed is the same as in Regnault's 

iment. This apparatus can be used to find the specific heal 

o£ ery high pressure and possesses the advantage that the 

gas is not wasted. 

For determining lhe specific heat at high temper;) he experi- 

ment of Holborn and Henningmay he mentioned. Employing -.u liable 
floe trie heaiers, resistance thermometers and well-designed calori- 

* If the i . ; relation po = RT/M (p. 5), where v i* the specific volume 

inn: M its molecular weij assumed to hold, a simpler expression 

For W ijned. Let V denote the volume of the reservoir 

1 a'szl/v, the density of the gas. Then 

W— V P = 3g£, and IY-W'=™ (p-f), 
where p is cxprtHsed in dynes and R in ergs. 



'RT 



MEASUREMENT OF C, 



51 



able to find the specific heats of nitrogen, carbon 
i team up to 1400°C. The calorimeter corrections are 

uncertain. The same method has been used by Nernst 
the specific heat of ammonia up to 600 °C. 

2ft. Constant-flow Method. For finding tire variation of specific 
ii i nperature the constant-flow method is most suitable, and 

used by Swan. The most recent form of the apparatus is 

thai used I !•. ill I'.' el and He use in finding the specific heat down to 

ow temperature and is in principle similar to that shown in 

1 to rider test., previously brought to a steady temperature 

ugh a suitable bath, flows through the calorimeter in 

i. Inside the calorimeter it is heau-'d electrically by 

in' roil of constantan and thus the energy supplied can be 

• I i M iv, of the incoming and outgoing gases 

ranee thermometers. The specific heat, can be 
v an equation similar to (15). 

li at low 1 mperatures the gas was initially passed through 

i| i rature bath in which the calorimeter was also immersed. 

'hum and other rare Scheel and lleuse modified 

o as to employ a closed circuit These experimenters 

1 icasurements on various gases in the range 0° to - 180 o C. 

Foi measurement of specific heat at high pressure the constant" 
od has been employed by Holborn and Jakob and gives 
ih able results. 






(b) MI'J HODS BASED ON THE MEASUREMENT OF C v 

.Steam Calorimeter. The direct determination of C. is best 
ol Joly's steam calorimeter (sec. 12). The method 



th< periment and the necessary details will be found 
on. 




1J - 1 ' i ■ • c |-l -i, Mm \ 




32 



CAI-ORIMKTRY 



[CHAP. 



30. Explosion Method* Following the work of Bunsen, Vieille 
and others, Pier improved the explosion method and devised the 
modern explosion bomb indicated in Fig. 13. Immersed in the water- 
bath B is a steel-bomb A which has a side tube M through which the. 
bomb can be evacuated and various gases introduced at the desired 
partial pressures. S is a corrugated steel membrane closing an open- 
ing in the bomb and carrying a mirror S. Light reflected from the 
mirror (alls on the photographic film F which revolves on a drum 3y 
applying various known static pressures and noting the deflection of 
the light spot, the pressure attained in any experiment can he found 
simply from the record on the film, Any explosion mixture, say, 
a mixture of hydrogen and oxygen together with tlit; inert ^as whose 
specific heat is required, is introduced in the bomb, L>y 'inert' is 
meant any gas which will not take part in the reaction either from 
want of chemical affinity or due to its presence in excess. The 
partial pressures of the various gases are known. The explosion is 
started by means of electric sparks and the final pressure reached 
is observed. This takes about .01 sec. 

The calculation may be easily made. If the vessel were allowed 
to cool to the initial temperature T 2 (absolute) , suppose it. would 
record the pressure pn. The maximum temperature T-, reached during 
the explosion is calculated from the value of p x , for, from^the gas 

, since the volume remains constant, 



r[ 



T — Pit 



Or ii ite the initial pressure and e. the ratio of the final to tb 

1 number of molecules (owing to the explosion the total m i 
of li 

= e Pi [ '" volume and temperature are me] 



■r —P* t 7" 



(28) 



where P is the ratio of explosion pressure to initial pressure. The 
relation connecting the specific heats and the heat of reaction is 

mih- (Tx-Tz) [mC ar -\-nC H ] f . . . (29) 

where m is the number of gramme-molecules of the reaction products ; 
n the number of gramme-molecules of the inert gas; C vr} C re ,resper- 
their mean molar specific heats over the range (T* - 7Y) , and Q 2 
the heat of reaction for the explosion mixture at T a °. C> £ is generally 
known from thermo-chemical data. We can determine C pr by explod- 
ing either with different amounts of a gas whose variation of specific 
heat with temperature is known, or with different quantities of argon, 
a substance whose specific heat is constant. Then the same reacting 
gases may be exploded with any inert gas, and knowing C v r , we can 
find C Di for the inert gas. 

The method is very suitable for measurements of specific heat 
at high temperatures and has been used to about 3000°C but suffers 
from the disadvantage that directly it gives only the average values 



H I 



DETERMINATION OF y 



58 



wide range and not the specific heat at any temperature. 

iluv, corrections are necessary for loss of heat, effects of dissoeia- 

n, incomplete combustion, etc. For argon. Pier found that the 

heat does not vary with temperature. 

3L list's vacuum-calorimeter method is employed to find 

ii low temperatures. The gas is enclosed in the calorimeter 

j. Eucken in this way found the specific heat of hydrogen 

. :>°K and obtained interesting results. 

METHODS BASED ON DETERMINATION OF y 

32. As already pointed out this is an indirect method of finding 

the s iecil c heats of gases. Though indirect it is capable of the highest 

v so that the modern accepted values of specific heats are 

on the values of y thus obtained. 

The methods for measuring y, the ratio of die two specific heats, 

ii !'• ie classified under two heads : (I) those depending on the 

■<ic expansion or compression of a gas, (2) those depending on 

the ty of sound in the gas. We shall first consider the former, 

(1) Adiabatic Expansion Method 
33- Experiments of Clement and Desoraws*— Clement and Desormes 

>: the first to find y by the adiab; tic expansion method. Their 

.inal apparatus has been considerably improved and is indicated 

in Fig. II, A large flask A of about 28 litres capacity is closed 

I • a stop-cock M about 1.4 cm. in diameter. Tlie flask is 

iiumected to the manometer P t -P,.[ by means of a side-tube and is 

with cotton wool to avoid loss of heat. First the flask is 

tialJy evacuated and the pressure p t recorded by the manometer, is 

observed. The stop-cock 

M is then opened 

and quickly closed. Air 

ics into the flask till 

i he -pressure inside and 

ide becomes equal, 

i 1 i ■: ■ process is adiabatic 

die loss of heat in 

the short interval for 

which the stop-cock M is 

• ii may be neglee 
The temperature of the 
air in the flask rises on 
unt of the inrush of* 
external air and the 
are becomes atmos- 
pheric. The flask is next 
allowed to cool to the 
imperature of the sur- 
• 'i idings when the water 
in I he :.■ On ter rises 
and finally indicates the ' 




Fig. 14.- -Clement and Desormes' appai 



- 1 ™ 






54 



GAL.OKIMETRY 



| GfiAPi 



Let the atmospheric pressure be p A and the specific volumes of air 
at the pressures, p- t p Ai p f , be respectively v it v A , i>/. The first: process 
is adiabatic and hence we have, assuming the gas lo be perfect, 

p z V-'---p.4l<A v TO 

Since the final temperature is the same as the initial, we have, 
considering 1 gram of the gas, 

SpFi^PfVf : 

Again, v A --v ; . (32) 

amse the volume of tile manometer tube is negligible compared with 
that of the flask and hence there will s no appreciable .. auge in the 
specific volume of the gas due of liquid in the manometer. 

Combining (30), {81), and (32), we have. 

Pa \pfh 

yJSiCJEiiL. 

log pi iogp f 
II, as usual j the changes in pressure arc small, 

' P-P, f 

From th irements of Clement and Disomies, Laplace deduced 

lu< of y to be IM 
In this experiment there is a source We have 

1 that the pressure inside has become atmospheric when the 
stopcock M is closed- Actually, however, osdllatio on 

'Lint of the kinetic energy more air first rushes in than would make 
pressure just atmospheric, and hence the pressure, inside becomes 
greater than p A . Next some air rushes out till the pressure inside « 
is less than p A and so on. After several such overshooting? the 
pressure p A h attained. This takes considerable time and, as a matter 
of fact, this to-and-fro motion has not subsided when t,he stop-cock 
is closed. It must be closed at the instant when during an oscillation 
the pressure just becomes atmospheric. This is very difficult to secure 
and nence later investigators tried to avoid it by measuring the change 
in temperature resulting from adiabatic expansion. The scop-coi 
has not to be closed in this case. We shall consider shortly the experi- 
ments of Lummer and Pringsheim and of Partington and Shilling 
baged on tills principle. 

We have assumed above that the incoming air has the same 
temperature as the air in the flask initially. To avoid correction in 
v it is not so, it Is better to start with compressed gas in the flask. 
when a U-tube manometer must be used in place of t^P^. Further 
care must be taken to use perfectly dry air lor y is appreciably 
different for moist air. Consequently sulphuric acid is generally used 
as the liquid in the manometer. 

54. Experiments of Joule, Lummer and Fringsheun, and Partington.' — 

Joule was the first to study the change in temperature by 




EXPERIMENTS OF "PARTINGTON 

Hllnbatic expansion or compression. Various investigators later 

i mi>!o\cd this method to determine y . Air was compressed m a vessel 

,1 it s temperature and pressure observed. It was then allowed to 



i In pressure and temperature before expansion, p 2i 
quantities after the expansion, we have, from (25) 



.[ suddenly to atmospheric pressure and the change in tempera- 
noted. Trie calculations can be easily made. If p tl 7\ denote 



T» the same 



l-v 



n 



(l-y) log 



! - 

" 



7", 



8^8 
Pa £ 1 

tog Pi- -1°$ Pi 



" Y (log Pi - tog Pi) - C lo S T i - J °g r a) 
y can be calculated, 
Lummer and Pringsheim made considerable improvements m the 
ratus for determining y by this method. They employed a 90-litre 
here and measured the change in temperature by the change 



topper sph... 

n i. sistance of a thin bolometer wire hanging atthe centre oJ this 

A Thomson galvanometer having a period ot 4 sec was 

Certain errors arc, however, inherent, in 



. . null 



instrument. 



1 1 ir apparatus 



ill llllll .1 LLia. . 

In order to eliminate these errors Partington has further improved 

pparatus. He used a large expansion vessel (130 litres capacity) 

: bolometer of very thin platinum wire (.001 to .002 mm. m dta- 



SfFI 




T 

PJ 5 ft 




\ j: 

a • \ii 






iftvwvvv— 

iv ■ .■•.-.- • - 



i 



/ 




Fig. 15.— Partington's Appafettis 
i witli compensating leads ; thus there was no lag. Further an 
♦This is true for a perfect gas only. For real gases it requires mmiifi cation. 



V 



56 



CALORUI' 



CHAP., 



Einthoven string galvunor pable of recording temperature in 

.01 sec. is used so that a detailed record of changes in tempera t Lire 
of the gas during and after expansion is obtained. His apparatus is 
indicaLed in Fig, 15, The vessel A is provided with the expansion 
valve- C which can be manipulated by means of the spring P and 
whose size can also be varied, A is connected to the sulphuric acid 
vi mometer M, the mercury manometer m, and the drying tubes F. 
Thus carefully purified air enters A. Further the vessel A is kpt 
immersed in a water-bath which is kept stirred by S. B is the bolo- 
meter wire (shown separately, in the figure^ and is connected In one 
arm of a Wheatstone bridge. G is the string galvanometer. 

The initial temperature was read on a car standardised 

mercury thermometer T immersed in the bath and was given correct 
to .01°. Then the resistance in one arm of the Wheatstone bridge 

lowered to give some deflection in the galvanometer. It was so 
arranged by trials that immediately alter expansion this deflection 
was reduced to zero. After the expansion experiment some ice was 
continuously added to the bath to keep its temperature constant and 
equal to than immediately after expansion. This was ascertained by 
keeping the galvanometer deflection steadily at zero, and the tem- 
perature of the bath was again read on the same mercury thermo- 
meter. 

.11' the aperture is too large, oscillations of the gas take place and 

•ilvanometer deflection is not quite steady, the initial deflection 
being somewh tcr than the true value. If the aperture is .too 

narrow, prolonged expansion resull be process is not adiabatic. 

Tn pr; rture was gradually diminished and when over- 

shooting was eliminated the deflection was instantaneous and per 
steady. The atmospheric pressure was read on a Fortin's barometer 
and y calculated from the foregoing formula, y was found to be 
1.4034* at 17°C. This method cannot be used at high temperatures 
since it is impossible to determine accurately the cooling correction. 

35. Ruchardt's Experiment. — A simple method for determining 
y, which is suitable for class-room demonstration, has been described 
by E. Riidiarrit, The apparatus consists of a Large glass bottle V 
(Fig. 16) fitted air-tight with a glass-tube at the top and a stop-cock 
H at the bottom. The glass-tube lias a very uniform bore in which 
a steel ball of mass m fits very accurately- If the ball is dropped into 
the lube, it begins to oscillate up and down and comes to rest after a 
few oscillations. If the period of oscillation be determined with a 
stop-watch, y can be easily calculated 

Let A be the cross-section of the glass tube, v die volume of the 
bottle, b the barometric pressure, and p the pressure in the flask. 
Then in the equilibrium position 



,-I + 3 






* Using this, the velocity of sound in dry air at 6"C was calculated fn a 
equation (3fi) to be 331.38 metres which is in close agreement with Hebb's mean 
value 331.41 m./sec. (Sec 37). 



1 



VELOCITY OF SOUND METHOD 



57 



ball now moves a distance x downwards it compresses the 
diabatically increasing the pressure to p + dp, hence ,tt equatum 



. diabatically 
ol motion is 

m - — Force of restitution 
= -Ad;>. 
pv* — constant, we have 



dv Ax 

■ypy-vPT- 



(35) 



i =-Ax. 
d*x 



s 




•ii .• lich the period 



of oscillation comes 



T = 2 



y = 



v, 

4ir •■• i 



(W) 



Fig. -16-— Rfidiardfs 
Apparatus. 



pA*T* m 

I litis, knowing T_, p and' the constants of the apparatus, y can be 
aked. 

(2) Velocity of Sound Method 
36 This method also depends upon the adiabatic expansion i 
• -,.'. , a gas but differs from the foregoing method in that 

direct measurement of changes in temperature or pressure need 
io be observed. The method has given us the most accurate data 
regarding specific heats Cor both high and low temperatures and so- 
all consider it in some detail. 
The velocity of sound in any Kind is given by the equation 
U — ^nrfe where E s p is the adiabatic elasticity of the fluid and 

density. For adiabatic changes in perfect gasesf pv v -. constant 
(p. 48) , hence 



Ef = - B ||fi} -tf tan (35). 



U 



V 



hp 



(38) 




Thus, if we determine the velocity of sound in the gas we car* 

37. We may adopt either of the jpo following methods. The 
;ite velocity of sound in the gas may be determined, or we may 

*See Barton, Sowd, , . . , ,_, 

ir real gases we must take into account the true equation of state [Lhap. 
c the value of \/.:pi'iiv), from that equation. In all accurate work 

d in. 












&s 



CALOHIMJE.TRY 






compare the velocity with that in another gas (say, air) which has 

been determined accurately by other methods. 

For our purpose we discard the large-scale determinations of the 
city of sound in air on account of the various defects inherent 

in them. The most, accurate direct determination of the velocity of 

; 'itcl in air- was made by Hebb in 1905 by a method depending on 

reflection of sound of known frequency from parabolic mirrors, 

His mean value after employing ail corrections gives 331-41 r 

sec, as the. velocity of sound in air at 0°G and 760 mm. pressure. 

Now we must remember that practically all determinations of the 
velocity of sound in gases have been made in tubes, but the velocity 
in ;:. tube is not the same as in free space. Corrections have to be 
applied to reduce this velocity to that in open space as explained in 
the next section. In equation (37) the velocity in open space must 

substituted. Dixon has directly determined the velocity of sound 
in different gases from 15°C to 1000°C in a very satisfactory manner. 

5 result may be employed to give y. 

Method based on the Measurement of Wavelength. 

3S. Kundt's Tube. — Kundt first devised an apparatus by means 
of which he could find the velocity of sound in a gas. This consists 
simply of a glass tube about 1 metre in length and diari eter. 

One end of the tube was fitted with a mo opper, while through 

b SrE s dl^l U^ 

17,—. Kundt's Double-tube A] 

loosely-fitting disc: carried by a glass or metal 
: ing-rod 11 clamped at its centre. Later Xundr 

employed the double-tube apparatus indicated in Fig. 17. Two tubes 
art; connected by means of the sounding-rod S which as 
; l] .- 1 at distances one-quarter of its length from either end. The 
.bber corks., d, d in the tubes A and B provide the damping 
ngement. The pistons P. P can be moved to and fro lo bring 
die tubes in resonance with the rod S. Throughout, the length oi 
each tube is spread some light dust such as Ivcb podium powder or 
silica dust. One tube is filled with air and the other with the experi- 
mental gas. The sounding-rod S is excited by rubbing it at the 
re when the dust is thrown into violent agitation" at the anti- 
nodes and collects at the nodes. The distance between successive 
nodes equals ha If- wavelength, and knowing the frequency of the sound 
the velocity is easily obtained. The double form of the apparatus 
*l very convenient for comparing die velocity of sound in anv ihts 
,'ith that in air for ' 




w 






. (39) 




VELOCITY OF SOUND METHOD 



59 



precautions are, however, necessary. The tube and the 

, us be perfectly dry. Carefully purified air must be used 

n for the various impurities must be made. Too much 

, not be used for excess of dust diminishes the ve loci y. 

in the diameter of the tube diminishes the velocity. 

m the velocity in the tube the velocity m open space must 

reed. Though mathematical equations giving the req m 

. on have been developed by Helmholtz, Kircbholt and others 

re not quite adequate and the best method is to express the 

U' in the tube as 

U' = U(l-kC), . . . ♦ (40) 

where V : velocity in open space, 

h -= a constant depending on the tube (its radius, thickness, 
the thermal conductivity, surface, frequency of the sound, 
etc.) 
C — a factor depending on the gas (its viscosity, density, ratio 
of specific heats, etc.) 
Kirchhoff showed that 



C = 



/fl'Ww)! 



here » = viscosity, p = density, Y r= ratio of specific heats and 






bte K being the thermal conductivity. For air from Hel ■■■ 

riments we know U } and by observing U[ and calculating C, k 

h tube is determined. This value of k is employed to give 
the veld-ay in open space for any other gas. 

Kundt and Warburg later employed this method to determine 
the velocity of sound in mercury vapour. One of the tubes contained 

ury vapour and was heated in an air-bath to about 800 C, The 
distance between two nodes was measured when the tube cooled. 
They found y— 1.666, Ramsay employed this method to find y for 
argon but on account of certain difficulties he got a low value. Behn 
and Geiger improved the apparatus considerably. They dispensed 
with the sounding rod and employed a sealed tube containing the 
experimental gas as the source of sound. This tube was clamped in 
its middle and was excited like the sounding-rod. The tube should 
be chosen properly and its length be adjustable so that the contained 
gas may give resonance with "the sound emitted by die rod. T he 
apparatus" was eminently suited for gases at high temperatures. This 
method was later>mployed by different investigators particularly by 
Partington and Shilling. 

* It can be easily shown from very simple considerations that for a mi* 
of perfect cases, 

_a_ = h + ii_ 

r— 1 yi— 1 72—1 
•where f h £s are the partial pressures^ of these gases, P the total pressure and Y 
stands for the ratio C P /C r for the mixture. 



60 



CALOR1METRY 



[chap, 

39. Experiments of Partington and Shilling. — These investigators 
determined the velocity of sound in various gases up to 10G0 D ~G bv 
a resonance method. The apparatus is diagram m a tically represented 

in Fig. 18. 

I'F is a silica tube 230 cm. long and wound over almost 

along its entire- length with heating coils. To this tube is attached 

it X a glass tube, MM, 150 cm. long. Inside the former is the piston 

1 J of silica carried by the rod A, also of silica. BB is a steel tube 

joined to A by means of a cork. The tube BB carries a saddl 




18.— Parting-tan and Shilling's ,'■• 

ing on a millimetre scale, thus die displacement of the piston 
Through this tube pass die thcimo couple leads to the 
potentiometer system £. The other end of the silica tube is closed by 
a telephone diaphragm T which can be moved by means of the screw 
Y. This end i losed gas-tight by means of the bell-jar J. . The 

diaphragm is excited by a valve oscillator V giving a note 
of Erequency 3000. D is a side-tube from which a rubber tube leads 
to the ear of the experimenter. X is ati asbestos phie to prevent 
radiation of heat to M. v 

The silica tube is filled with the experimental gas and maintained 
at Che desired temperature. The central tube AB is gradually moved 
• from 1} and the successive positions of the saddle I on the 
millimetre scale corresponding to a maximum sound in D are noted. 
The successive distances correspond to A/2 and knowing the frequency 
the velocity is determined. The position of T has Co be adjusted at 
different temperatures in order to give maximum sound in D when it 
will be at a distance A/2 from the "latter. 

nno ?°°k employed this method to find y for air and oxygen from 
SO to p3 £. His apparatus may be visualized if we imagine the 
hot-air bath of Kundt and Warburg to be r-.huvd Uv :i h; ...- 
flask containing liquid air. 




specific; heat of superheated vapour 



01 



■ i 1 1 1 i 



40. Specific heat of superheated or non-saturated vapour.— Reg- 

_.£ determined the specific heat of superheated or non-saturated 

id other vapours with an apparatus which wa itially 

liar to that shown in Fig. 12 (p. 49). Steam is superheated to 

by passing it through the spiral S in the oil-bath kept at a 

.Lure above I06°C, The superheated steam is next passed 

n constant pressure into the condenser kept immersed in the water 

trimeter, and the rise in the temperature of the latter from 8 X to a 

noted. 

If m h the mass of steam condensed, e p its mean specific heat 
. rant pressure between the temperature T and its condensing 
tnt 0° at constant pressure, and L the latent heat of steam at d°> 
the quantity of heat given by the steam is 

Ulcers) +mL+m{&-8J, 
This must equal a; (0s- fli) where w is the thermal capacity of the 
lori meter and its contents. Hence 

]T-B)+mL+m{e-6 s )=w{6 s -e t ). . . (41) 

experiment is then repeated with another value of T and a 

►n< ' n similar to the above obtained. Solving the two 

qu the wo unknowns c p and L are determined, Regnault 

thus found c t for steam between 225*C to 125 L 'C to be 0.48. 

In case of vapours of other liquids, the specific heat of the liquid 
must be taken into account in writing out the above equations. 

41. Results.*— In the foregoing pages we have considered the 
various methods of finding specific heats. In tablef 5 (p. S2) we 

e the values which best represent Lire experimental data. We 
reserve our comments on these values for the next chapter. 



replaced by a long Dewar 



| Taken from TLmdhuch tier E.vpfn>r<e> k, Vol. 8, p. 33. 



62 



CAI-ORIMETRY 



[chap. 



Table 5. —Molar Heats in Calories at 20 °C and 
Atmospheric Pressure, 



Gas 


Cp 


e. 




Remarks 


Argon 

Helium 


4.P7 


2,98 


1,666 


> 


4.97 


2,98 


1,666 


t Monatomic 


tlrogen 


6.865 


4,88 


1,408 


1 


Oxygen 


7.03 


5,035 


1.396 


Nitroj 


5.93 


4.955 


1.402 


| 


Nitric oxide 


7.10 


5,1.0 


1.39 


-Diatomic 


Hydrochloric acid , . 


7.04 


5,00 


1.41 




Carbon monoxide . . 


6.97 


4.98 


1,40 




Chlorine 


8,29 


6,15 


1.35 




Au 


6.950 


4.955 


1.402 




Carbon dioxide 


8,83 


6,80 


1.299 


I 


Sulphur dioxide . , 
Hydrogen sulphide 


9.65 


7,50 


1.29 


- Triatomjc 


8.3 


6.2 


1.34 




. 


8,50 


6,50 


1.9] 


\ 


i' ili 


12.355 


10,30 


1.20 




lene 
ene 


10.45 
10.25 


8,40 

8,20 


1,24 
1.125 


r Polyatomic 




8.80 


*&! 


1.315 





42. Special Calorimeters.— Various types of calorimeters have been 
devised for special purposes e.g., for the measurement of the heat 
of coin bastion, heat of chemical reaction, heat of dilution, etc., but 
they involve no new principles. Particular interest, however, attaches 
to the determination of heat of combustion in industries, for the 
value of fuel u judged mainly from its calorific value. This heat 
can be easily determined with the help of the calorimetric 'bomb'. 
Fig, 19 indicates Che calorimetric 'bomb*. It consists of a stout 
el-cylinder A fitted with a cover held down tightly bv suitable 
means. The cover has a milled-head screw valve which varies the 
cavity A a and thereby regulates the admission of oxygen through the- 
tubes B and C into the bomb. Through the centre of the cover but 
insulated from it passes the wire i which is connected to die platinum 
wire wj the other end of the latter being connected to e. There is 
another similar screw valve varying the cavity k, through which -as 
«] go out of the bomb. To enable the bomb to withstand the 
osive action of the products of combustion it is plated inside with 
gold, though platinum would be better. The bomb is enclosed in an 




HEAT BALANCE IN THE HUMAN BODY 







calorimeter such as is used for the method of mix! 
ilorimeter is provided with a 
and accurate mercury thermo- 
. i . I he whole is surrounded by 

n.l temperature jacket. 
To find the heat capacity of the 

I. Ir and its accessories a known 

mnt of electrical energy may be 

iii in the system or a fuel ol 

calorific' value burnt. The 

method is adopted in 

ing laboratories and the 

t in actual practice. Benzoic 

id is most suitable* for this cali- 

bra t ion. The fue 1, if solid, is formed 

i ;all briquette; if liquid, it is 

ul i d in pure cellulose and put in 

I he platinum dish F and ignited. 

mi three times the amount of 

just necessary For complete 

combustion Is admitted through B, 

l"' oxygen is generally employ- 

cd ;n a pressure of about 25 atmos- 

and at this high pressure the 

obustion is almost instantaneous, 

,i result of these experiments it 

en found that the calorific 

i »f anthracite coal, wood (pine), 

Lrol and methylated spirit are 8.8, Fig. 19— The Calorimetric Bomb-. 

i.l, 11.3 and 6.4 kilocalories per gram respectively. 

43. Heat balance in the human body. — The temperature of the 

human body remains almost constant in health. The chief loss ol' 

heat is from the skin and from the excretory functions, the skin 

niracting in winter to diminish this loss. Heat is supplied to the 

ly by the food we eat and by the oxidation of living tissues and 

i ides. The blood stream serves to keep the temperature of the 

ody uniform. Due to the larger heat loss from the body in winter 

have to take more food and cover ourselves with heavier clothes. 

Books Recommended 

!. Glazebrook, A Dictionary of Applied Physics } Vol. I, article 
on 'Calorlmei 

Partington and Shilling, Specific Heats of Gases*. 

3. Handbttch der E^perhnenlolphysih, Vol. 8, Part T. 




CHATTER HI 
KINETIC THEORY OF MATTER 

The Nature of Heat 

1 Historical,— In the legends of some ancient nations, it is 
iade for man by some friendly spirit by rubbing 

-re bis- 




increase 
be highly 
any process 



the c-ai-lv~philosophers had no correct notion aooui iu ^J p£*lo- 
"ophS-d I oin the observation that heat could pas, spontaneously from 
a C body to a cold one. Heat was, thereto-, supposed to be a 
kind oE fluid— the caloric ftutd. 

Various fictitious properties were assigned to this JOT*™"" 
fluid. It was supposed lo possess no weight, since bodies did not 
rrLe in weigh/ on mere heating. Further, it was supposed u> 
elastic all-pervading, indestructible and uncreatabk by 
' The particle., of this fluid were supposed to repel one 
another strongly which explained the expansion of bodies when heated 
d l" o tlielniission of Seat during combustion. Temperature wi 
, ;,r to potential or level. When the body was heated the caloric 
id was supposed to stand at a higher level than when cold Pro- 
of heat bv friction was compared to the oozing out of water 
™ m a sp , Th e caloric fluid when thus squeezed 

out n I itself as i 

Doi caloric theory of heat began to be thrown cowards 

Leenth century. The earliest philosopher to have 
. ,.u:al nature o£ heat was Count RumfonL* 
In those da a:rc made by casting solid cylindrical preces and 

inside by a boring machine, Rumford in 1798 
hat apparently an inexhaustible amount ot heat could be 
prod the friction of the spindle o£ the boring machine against 

body of 'the gun, though the amount of iron scraped was very small 
lie undertook protracted experiments and found that the amount 
of heat produced (measured by the raising of water to a high tem- 
perature) bore no relation to the amount of iron scraped, but was 
proportional to die amount, of motion lost. He henceforth rejected 
the caloric theory and asserted that Heat is only a kmd of Motion, 
Whenever Motion disappears it reappears as Heat and there js an 
£xact proportionality between the two. He even made an estimate 
of what we now tali the Mechanical Equivalent of Heat. His value 
: not much different from the value now adopted as standard. 

* Count Rumiord (1753-1814) was born in North America Rein r loyal to 
Grr , .., | i:r [ :T: , the American War of Independence, he had to flee from 

bis country. He entered into the service of the Prince of Bavaria aiui 

I in charge of the arsenal at Munich when he perforated the ;.-■:•!, vatec 
••-, die bonus of guns. Tn 1799 he went to London and was one 
of die Sounders of the Royal Institution, 



Face \h 64, 




James Pkescott Joule (1818— 1880) 



(P- fc) 



Born near Manchester, Joule was educated at home. The main work 

which occupied the greater part of his life was on the relationship 

between work and heat. He established the principle 

of the mechanical equivalent of heat 







James Cleek Maxwell (1831— 



Born in Edinburgh, educated at Edinburgh and Cambridge. He became 

Km* a Colkge, London, in i860 and Professor of Experi- 
mental P hy5fcs ■ charffe of ^^ ,£££* 

Abridge, «n 1871. Hi, greatest work was in con- 
necbon with the development of the kinetic 
theory of gases and the foundation of the 
electromagnetic theory of light. 



(P. 75) 




FIRST LAW Of THKRMODVNAM1CS 



65 



In 1799. Davy showed that when two pieces of ice were rubbed 
gether water is produced. It was admitted by all that water lias 
greater quantity of heat than ice. I Now supporters of the caloric 
; i v asserted that heat is generated in friction because the substance 
produced by friction has less capacity for heat than the original subs- 
tance. But the substance produced in Davy's experiment (water) 
has greater heat capacity than ice, hence the caloric theory became 
untenable. Davy's experiment proved the greatest stumbling block 
for the caloric theory. 

But the valuable work o£ Rumford and Davy was soon forgotten 
nd it was *■- njy about forty years later I hat the first law of Thermo- 
dynamics gained general publicity through the researches of Joule in 
'•land, M and ; Lmholtz in Germany, and Colding in 

Denmark. 

2. Joule's Experiments, — In 1840 J, Joule of Manchester began 
his classical experiments for determining the relation between the work 

ne and herd generated. We do not wish to describe these experi- 
!i detail as they, are at present only of historical interest 
rhe heal was produced by churning water contained in a cylinder 
bv means ol brass paddles. This could be kept revolving by means 
• l a chm I ilc thread wound over a solid cylinder and passing over 
pulJe ' rying weights at either end. The amount of work 

i lated by observing the height Lhrough which the weights 
fell, i ii' rise in temperature was measured by, a mercury thermo- 
meter and hence the heat generated could be found. After app] 
'.hi ections Joule found that 772 ft.-lbs. of work at Manchester 

the temperature of 1 pound of water I°F. In 1878 Joule 
used ;i modified form of the apparatus in which the work done was 
■ the application of an external couple as in Rowland' 
to be described later (§5.) 

3. First Law of Thermodynamics, — The conversion of work into 
i , thus established by Lhe experiments of Rumford and Joule. 

e ] the times of Rumford and Joule, the steam-engine 

n widely applied for various industrial purposes. As we shall see 

, this is simply a contrivance for the conversion of heat into work. 

Thus it is established that heat and work are mutually convertible. 

. ■ may go even further and say that when some work is spent 

ng heat, a definite relation exists between the work spent 

1 n era ted. These two facts, viz., the possibility of converting 

to heat and vice versa and the existence of a definite relation 

"ii the two are expressed by the First Law of Thermodynamics. 

Matin itii illy, the law may be seated thus: — 

If IV is the work done in generating an amount of heat H, 
we tii 

W = JH, I: 

t This was proved by Black'.? discovery of Latent Heat of Fusion. He 
equal masses of ice and ice-cold water alternately inside a room and 
observed that the water was raised 4"C in half-an-ftour, while the ice uiok about 
ten hours to melt into water, the temperature remaining constant. 

5 



—7- 



66 



KINETIC THEORY OF MATTKU. 



: 



where / is a constant, provided all the work done is spent in p; i 
hie heat and no portion is wasted by fricdon, radiation, etc. If H 
U expressed in calories and W in ergs, J= 4.186 X W- The truth 
of the second statement embodied in the law is amply proved by the 
fact that Lhe various methods for finding / (sees. 4—7) yield almost 
identical values, 

4. Methods for determining ./.—Various methods have been d< 

ed for finding the value of the mechanical q > of heat hut 

the method of fluid friction and the electrical method are the only 
ones capable of yielding accurate results, and hence only these will 
be considered in 'detail. There is, however, an ingenious method of 
calculating the value of / which was first given by J. R. Mayer in 
1843. From the theoretical relation J(G P - C v ) — R (p. 46), he cal- 
culated the value of /. Thus, Eor hydrogen R = p*VJT» — 8.3 14 X W 1 
ergs per mol per °C (pp. 6, 10) and C p -C,= 1.985 cal. per mol per 
°C (p. 62) . 

8*314xlQ 7 
.*. J— — -1--985 ~ = 4 - T8 X 1{)7 er S s P er calorie. 

Certain other methods* that have been employed are enumerated 

: 

(I) Measurement of heat produced by compressing a gas— Joule. 
Heat produced by percussion — Him, 

(3) Work done by a steam-engii 

This wi Him in 1862. He measured the amount 

the cylinder of the steam-engine in a 

at a known temperature and pressure. The 

total heal y the engine was found by conducting 

ill- >• j i>: steam into a calorimeter, and the heat loss due 

to ci r causes was estimated. Thus the net 

"••■Lint of heal; which is converted into work is obtai I 

ik done by the engine was found from an indicator 

din;; ram (CI ap. IX.). Equating these two Hirn got a 

value of .7 = 4.18 X 1° 7 ergs P er calorie. 

(4) Heat developed in a cylinder kept stationary in a rotating 

magnetic field produced by means of p ilyph: ie alternating 
I ictric current — Bailie and Ferry, 

5. Rowland** Experiments.— Joule's thermometers were not :- 
dardised and thus errors of 1 or 2% may arise from this cause. Tiie 
rate of rise of temperature in his experiment was rather slow (about 

1 per hour) and hence the radiation correction was : 
Rowland minimised this source of uncertainty by designing a special 
apparatus with the object of securing a rapid rate of rise of tempera- 
ture (40°C per hour) , the principle of the method being identical 
with that of Joule. 

♦ For a ' '.'-.l a$ Llit: methods see Glazebrook, A Dictionary of Api 

,. V ■!. I, p. 480. 




in 



ROWI -AND S EXPERIM ENTS 



67 



'I fur calorimeter was firmly attached to a vertical shaft ab [Fig. 
ch is fixed a wheel kl wound round with a string carrying 
;it either end, the whole being suspended by a torsion 






- 



A. 




Fig. 1 (a). — Rowland's Apparatus. 

wire. The axis of the paddle [Fig. 1 (b) ] passed through the bottom 
of the calorimeter and was attached to the shaft ef. The latter could 
>tated uniformly by the wheel g driven by a steam-engine. The 
number of revolutions jjyas automatically recorded on a chronograph 
worked by a screw on the shaft ef. The revolution of the paddle at 
an enormous rate tended to rotate the calorimeter in the same direc- 
tion on account of fluid friction. This was prevented by the external 
couple produced by thi ■ <;. p and the torsion wire. For the 

' urpose of accurately determining- the radiation correction a water 
|acket tii surrounded the calorimeter. 

The paddle is indicated separately in Fig. 1 (c). To a hollow 
cylindrical axis four rings were attached, each having eight vanes. 



68 



KINETIC THEORY Or MATTER 



CHAP. 



Around these were the fixed vanes, consisting of five rows of ten 
each, -which were fixed to the calorimeter. Thus the liquid could 




T£< 







Fig. 1 (b). 



F3g.l(tf). 



b* vigorously churned. The rise of temperature was recorded by a 

thermometer suspended within the central sieve-like cylinder in which 
water circulated briskly. 

If D denotes the diameter of the torsion wheel and mg, mg, the 
ded, the work W done in » revolutions of the paddle 

is given by 

W — couple X angle of fcwist — ffigD- 2/ttfi. . , {*•) 

the thermal capacity of the calorimeter and its contents, 
perature i Ldiation) , the heat produced 

by fri MB8, hence 

(3) 



2mt. tngD 

3 — MBd~ 



If D is in an, and mg in dynes, / conies out in ergs per calorie. 

Corrections were applied for the torsional couple, for the weights 
in air which must be reduced to vacuum, for expansion of the 

til wheel, etc. Rowland found J =4,179 X lu? ergs for the 20° 
calorie. La by recalculated from Rowland's observations by applying 
corrections and obtained the value / = 4.187 X 1° T er g s * or ^ 15 ° 
calorie. 

Reynolds and Moorby obtained bv a modified apparatus, the value 
of the mean calorie between 0° and 100 *C to be 4.1833 X 1 T ergs. 
Hercus and Laby employed what is in principle an induction motor, 
to find J, and obtained the value / = 4.186 X *0 7 ergs per caloric. 

6. A Simple Laboratory Method of finding J.— For laboratory pur- 

a simple apparatus for finding/ due to G. F. C. Searle is 

in Fig. 2." but the accuracy, attained by this apparatus 

is not great. A is a brass cone held "rigidly in position by means of 



in. 



ELECT] "R I £ : AL METHODS 



69 




Fig. 2, — A simple laboratory apparatus 
for /, 



non-conducting ebonite pieces attached to a brass cylinder C, which 
am be made to revolve by means of a motor. Inside A is another 
; cone B fitting smoothly into it and attached rigidly to a wooden 

disc D. The latter has a groove 
running round its circumfer- 
ence and carrying a cord which 
passes over the pulley P and 
supports a weight mg. When 
die outer cone rotates rapidly 
die inner one tends to move in 
the same direction on account 
of the friction between the two 
cones, but is held in position by 
properly suspending a suitable 
weight mg at the end of the 
cord. Tire inner cone B con- 
tains some water, a thermo- 
meter and a stirrer (not shown) . 

When the weight mg ^ is 
kept stationary the turning 
moment exerted by it just, bal- 
ances the frictional couple. If 
D is the diameter of the disc 
the frictional couple is mgD/2 
and die work done by it in n 
revolutions of the cone is 2nir.mgD/2. If M be the water equi- 
valent of the cones and the contents, $0 the rise of temperature pro- 
duced by friction, then 

nirmgD — JmlQ, . , - . (4) 

whence / can be calculated. 

7, Electrical Methods.— These methods have already been des- 
cribed (pp. 37-41) in full detail. It is easily seen that if the specific 
heat of the liquid be known, this method gives the mechanical equi- 
valent of heat. There are tw r o methods: — (1) Steady-flow method, 
(2) Rise in temperature method. The former was employed by 
Callendar and Barnes who did their experiments with great care and 
skill but the principal source of uncertainty in their work lies in the 
value of the electrical units employed. According to Laby we may 
put the E.M.F. of the Clark cell used by Callendar and Barnes as 
1.4335 volts at 1&°C". Tlie*y employed the international ohrn which is 
equal to 1.0005 X 10° e.m. units. Reducing Calendar's results with 
the help of these data Laby gets for the mechanical equivalent of 

20° calorie a value of 4.1795 X 10 7 ergs. This yields 4.1845 X J0J 
ergs as the equivalent oE the 15° calorie. The most accurate experi- 
ments on the subject are those of jaeger and Steinwehr by the rise 
of temperature method, and of Laby and Hercus by the mechanical 
method, the respective values being 4.1863 X W a nd 4.1860 X 10T for 
the 15° calorie, Osborne, Stimson and Ginnings have recently found 



70 



KINETIC THfcORY Ol? MATTER 



CHAP, 






i In: value -1.1858 by the electrical method. Hence, we can adopt the 
value 4,186 >< 10 r ergs P er calorie. 

Exercise. — Joule found that 778 li.-lbs. of work can raise the 
mperature of 1 pound of water IT'. Calculate the mechanical 
equivalent of heat in C.G.8. units. 

778 f L. lbs. =; 778 )< 30.48 )< 453.6 gm. X cm. 

= 778 X 30.48 X £53.6 X 981 == l-«55 X 10" ergs. 
Heat produced = 453.6 X $ = 252 calorie.s. 
,\ J =z 1.055 X 10i*/252 = 4.187 X 10 T ergs per calorie. 

8. Heat as Motion of Molecules. — From these experiments it is 
[early established that heat is a kind of motion; the- next question is 

—motion of what ? The answer was given by Clausius and Kronig in 
1857 for the case oL gases. They said thai onsists in the motion 

of molecules or the smallest particles of matter. 

The idea that if we go on dividing matter we ultimately come to 
imall particles which cannot be further sub-divided dates from very 
ancient times, But it remained a barren speculation till Dalton gave 
to it a definite form in the middle of the last century. The history 
of the molecular theory is known lo all our read md n< i ount 
Et need be given here. that according to it all 

1 oE a large tiumbei of molecules, all molecules of 
Line substance being exactly idem regards ■•, etc. 

ii. En the solid and liquid states these mole- 
while in gases they are Ear apart from one 
her. 

to i lie Kinetic Theory of Matter hear, is supposed 

to consist in the motion of tin ecules. The identification ol heat 

with motion of molecules is not a mere hypothesis. It is able to 

explain and predict natural phenomena and at present there is little 

i solid foundations of truth. 

9. Growth of the Kinetic Theory. — The Kinetic Theory of matter 

two fundamental hypotheses : (1) the molecular structure 
ol matter, (2) the identification of heat with molecular motion. The 
1 1 'St of these was established early in the 19th century while the second 
established by the experiments of Rumford, Joule, Mayer and 
Colding. We may, however, consider Daniel Bernoulli! (1730) as the 
founder of the modern kinetic theory as he was the first to explain 
Boyle's law by molecular motions. Clausius and Maxwell in the 
middle of the 19th century placed the theory on a lirm mathematical 
basis. Among the other prominent contributors to the theory are 
Boltzmann, Meyer, Jeans, van der Waals, Lorentz and Lord Rayleigh. 

Up to the beginning ol the present century, however, the theory 

had been developed entirely from a mathematical stand-point. There 

no direct experimental proof of the actual existence of these mole- 

les or of their motions. Gradually, however, much evidence has 

•Since the discovery of isotopes tlii: remark requires some modification. 



in. 



EVIDENCE OF MOTJttL'LAR AGITATION 



71 



accumulated in favour of these views, the most important being the 
Brownian Movement phenomena investigated by Pcrrin in IW&. 

10. Evidence of Molecular Agitation.- (1) The phenomena of 

ssrtsssssse: & 

nde o C0 2 . Alcohol 

„.,. d i ffus es into the entire mass or : , 

water E«n sold is found to diffuse into leaO. 

k) A jeu tends to expand. It is a common 
experience that the bh ufcs of the gas 

'd To dv away and FT^jZ 

ansibility. T& rectilinear motion ol the mole- 
culef is 'ruled by the experiment of Dunoyer in 
w hi, obtained atomic or molecular y;v 

t™ annus consists of the tube ABC (Tig. 

I" u Compartments, A, B, C by p *** 

ii-mhri«ns 1> and F-. The apparatus is highly e\a- 

SS2 *fr ic .idc-Bibe J. The end F contain- 

£ heated to abou Ift^S 

* through circular apertures in 1 and 

deposited on . tamd J f ] ; 

the second hole, form . ' h ^ 

holes bv straight lines, which proves that the mole- 
1 : straight lines. 
. Phenomena of evaporation and vapour 

14')' The m s I aw s ca n 1 >c dedu red from tli e kinetic 
theory'". . 13). Other results ob from the 

-, as suecihe heat, iher 

molecular diameter, etc., agree with experimental results or deductions 
from other branches of Physics. 

(5) Phenomena of Brownian movement. 

11. Brownian Movement.*- This phenomenon was first discover- 
cd bv the English botanist brown in 1827 while _ ol 
aqueous suspensions of fine inanimate spo u der high powei 
,pc. He found the spires dancrng about in the wild 
I, ion. The phenomenon can be readily observed d small particles 
ided in ; as in a colloidal solution, are examined n 

, ■ ■ ... __ u A *. n » .. n Jn» 'in iiUr^.mirrosrnTkf. the 



A 



F 



l-\ . " - 
Dtmoyer's 
apparatus. 




■■■ f, [- detail di ' siom see the Authors" A Treatise or. Heet, |3.4S— 3.4S, 



72 



KINETIC THKORY OF MATTER 



CHAP. 




V/ 



I Vilhittjih^J" 



The Browniari movement never ceases -it is eternal and spon- 
taneous, and is independent of the chemical nature of the suspended 
panicles, all particles of the same size being equally agitated. Smaller 
particles are, however, much more vigorously agitated than bigger 
ones. The motion becomes more vigorous when the temperature is 

aaed or a less viscous liquid is chosen. It is just perceptible in 
glycerine and very active in gases. No two particles are found to 
execute the same motion, hence the motion cannot be due to any 
convection or eddy currents. 

Th,e discovery of such spontaneous motion, and the fact that the 
motion is maintained even in viscous liquids without die application 
Oi my force was a great puzzle to earlier observers. Gradually, how- 
ever, it has been established that the Brow ni an movement is due 
to the impact of the surrounding molecules of die liquid on the 
Brown jan panicle. It is evident that the 
Aficroscoj* forces due to molecular impact, will almost 
completely balance if the size of the particle 
is very large (say, a large body immersed 
in the liquid) but there can be no balance 
if the size is small Any small particle will, 
therefore, be acted on by a resultant un- 
balanced force .and will consequently exe- 
cute motion. As this force varies at 
random, so the motion of the particle "'.'■ 
be at random and will be some*' 
Fi K 6 (p. 81). Thus the phenomenon 
aent is a d of the existence of w i 

ion. 

The study of the kinetic theory is best approached through a 
ases. The kinetic theory of the liquid and solid states is 
comparatively undeveloped and will not be discussed m this book. 

12. Pressure Exerted by a Perfect Gas.— It has been shown above 

that a gas consists of molecules in motion. As a consequence, it 

wre on the walls of its enclosure. To calculate this 

pressure we first make several simplifying assumptions. These are 

the following ;. — . 

(1) Though the molecules are incessantly colliding against one 
mother and having their velocities altered in direction and magnitude 

each collision, yet in the steady state the collisions do not affect 
the molecular density of the gas. The molecules do not collect at 
one place in larger numbers than at another. Further, in every 
1 nient of volume of the gas the molecules are moving in all direc- 
tions with all possible velocities. The gas is then said to be in a 
state of molecular chaos, 

(2) Between two collisions a molecule moves in a straight line 
with uniform velocity. This is because the molecules are material 
bodies and must obey the laws of motion. 



Emufok 
viug 



ill. 



PRiiSSL'RK EXERTED 1SY PERFECT GAS 



7$ 



(3) The dimensions* of a molecule may be neglected in com- 
parison with the distance traversed by it between two collisions, called 
its free path. The perfect gas theory- treats the molecules as mere 
mass-points. 

1 1 The time during which a collision lasts is negligible com- 
pared' with the time required to traverse the free path. 

(5) JThe molecules arc perfectly elasticf spheres. Further, no 
appreciable force of attraction or repulsion is exerted by them on 
one another or on the walls, i.e, ? all energy is kinetic. This is proved 
by Joule's experiment (p. 47). 

We now proceed to calculate the pressure exerted by such a 
gas. We will employ the method of collisions, because it is very 
simple. 

Imagine a perfect gas enclosed in a cube of unit sides and con- 
sider a molecule moving with the resultant velocity c and component 
velocities «, v, w along OX, OY, OZ axes respectively. " The axes 
are taken to be parallel to die sides of the cube. The molecule 
collides with the surface of the cube perpendicular to OX with the 
velocity u. From the principles of conservation of energy and 
momentum it follows that it will rebound with the same velocity. 
Hence the change in momentum suffered by the molecule during 
collision is %mu. The molecule strikes that particular surface «/2 
times per second, hence the change of momentum per second is 
2imi.u/2 = ?nu s . Since pressure is equal to change in momentum 
per second, the total pressure exerted on that surface is "Zmu 2 where 
the summation extends over all the molecules. 



/. p x — Xmu*. 



• (5> 



Now "'— = w" J where u- is the average of u 2 over all the n 

molecules. 

,\ p K — mnu 2 ..... (6) 
Similarly, the pressures on the other surfaces are 
p y = mntJ 2 , p z = mnwK 

*If we consider the dimensions of the molecule and the forces of attraction, 
we gel van der Waals' equation (Chapter IV). 

t The assumption of perfectly elastic collisions, on the average, is warranted 
by the fact that we can convert into work all the heat supplied tr> a perfect 
gas. For otherwise addition of heat would increase molecular velocities ami 
hence also the force of collision, and if deformation of molecule results, all beat 
may not be converted back to work. The picture here given is essentially that 
of a monatomjc molecule; there will occur deformations of polyatomic molecules, 
accompanied by an increase or decrease of rotational and vibrational ci 
but on the average there is no net los3 or gain of translational energy during 
collision. Equation (9), however, can be deduced without the assumption of 
perfectly elastic collision. For details see the Authors' Treatise on Heat, Sec. 
3.12, footnote 2. 



74 



KINETIC THEORY OF MATTER 



': CHAP, 



where a» a , ar 2 denote the mean square velocities in the other two 
perpendicular directions. Since experiments* show that p x= p^=^&, 
we have 

u % z=^iP —~w2. n\ 

This is also to be expected from the fact that the molecules do not 
tend to accumulate in any part of the vessel. But 

n n 

where c 2 is the resultant mean square velocity. Hence from (7) 
and (8) 

IP = £^ 
and finally (6) yields 

p = $mnc*. (9) 

But mn = Ps the density o£ the gas, since ra is the number 
per ex. 



p = y <>\ 



(10) 

(11) 



where £ is the kinetic energy per unit volume. Thus we see that 
the pr. a perfect gas is numerically equal to two-thirds of 

kinetic energy- of translation per unit volume. 

ie molecule as if it suffered no collision 
i others. If a molecule encounters another it gives to 

ther which bv ;, h as t0 traverfle 

tance as the first one if there had been no 
tion (■!) no time is lost in this collision and 
ur calculation holds true for a perfect <ras. 

(10) enables us to calculate the' mean squ 
Kity c" of the molecules of any gas, for 1?=-. $pf p . The pressure 
;1 " ; of a gas can be found experimentally and hence c= 

calculated. Tims the density of nitrogen at 0°C and atmosp! 
pressure is 000125 gm. per cc Hence for nitrogen the root me 
square velocity! 



C = 



V*- \/- 



3 X 76x13-59x981 iM 

rjm - = 4.98X10* an./sec. 



*Thk is so for very email cubes. Rigorously speaking, for large cubes 

tht SSL " "* to k ***** m aC ^ of sSvitation^T ptS on 

is^mult may b c easily deduced from the laws of elastic impact 

? C is the square root of the mean square velocity and differs from the 

m and 20 i s Ta whale the root mean square of 10 and 20 is 
/HP +20* _ . 
V 2*~" =1S ' 81 ' Thi: exact relation between C and c for the molecules 

of a gas in equilibrium h Riven later (§16), 



ill. 



DEDUCTION OF GAS LAWS 



75 



The formula also shows that the molecules of the lightest gas, 
,, hydrogen, would move faster than the molecules of any, other 
gas under the same conditions. 

13. Deduction of Gas Laws— From the above results we proceed 
to deduce the laws of perfect gases. 

(1) Avogadro's* Law, — If there are two gases at the same 
lire p vflft have from (9) 

p = im i n 1 C t a — sJm 2 n 2 C/., . . . . (12) 
where the subscripts 1 and 2 refer to the first and the second gas 
respectively. Further, if the two gases are also at the same tempera- 
ture we know there will be no transfer of heat (or energy, since the 
two are equivalent by the First Law of Thermodynamics) from one 
to the other when ihey are mixed up. On mixing, the two types of 
molecules will collide against one another and there will be a mutual 
sharing of energy. Maxwellt showed purely from dynamical con- 
siderations- that the condition tor no resultant transfer of energy from 
one type of molecules to the other is that the mean translation a] 
energy of molecules of the one type is equal to that of the other. 
Hence if the two gases are at the same temperature it follows that 

}*&* = &#* (13) 

Combining (12) and (T3) we get 

n^=n 2 , (14) 

I.e., two gases at the same temperature and pressure contain the 
same number of molecules per c.c. This is Avoga tiro's Law. 

(2) Boyle's Law. — Equation (10) states that the pressure of a 
,;i-. is directly proportional to its density or inversely proportional 

to its volume. This is Boyle's Law. This holds provided C a remains 
constant which, as shown above, implies that the temperature remains 

constant. 

(3) Dq Hon' s Law. — Tf a number of gases of densities p ti p.p p% 
... and having mean square velocities Ci 2 , tV 2 , C 3 2 ....,. be mixed' in 
the same volume, the resultant pressure p } considering each set of 
molecules is given by 

p --- taG 1 % +bA*+WV+' ■ 

— Pl+P2 + P*+— .♦►♦». (15) 

♦Count Amedeo, Avogadro di Quarcgua (1775 — 1856) was born in Turin 
where he was Professor of Physics from 1833 to 1850. His chief work k 
Avogadro's law, 

t Maxwell coiiHidcred the collision of gaseous particles of two different 
-■ and possessing different amounts of energy. By applying the dynamical 
laws of impact viz. conservation of momentum and energy, he found that after 
each collision the difference in energy oi the two molecules diminishes by a 
certain _ fraction, i.e. the molecule possessing greater energy loses it while that 
possessing less energy gains, This process is repeated at each collision, and ulti- 
mately the energies of the two become equal. For details, sec the Authors' 
Treatise on Heat (1958), pp. 845^847. 



76 



KINETIC THEORY OF MAI EER 



CHAP. 



ITT.] 



MAXWELL S LAW OF DISTRIBUTTON OF VELOCITIES 



77 



i.e tf the pressure exerted by the mixture is equal to the sum of the 
ires exerted separately by its several components. This ia 
Dalton's law of partial pressures. 

14. Introduction of Temperature. — If we consider a grain- 
molecule ol the gas which occupies a volume V, equation (10) fields 

pF=z$MC 2 , (16) 

M being the molecular weight. In order to introduce temperature 
in the foregoing kinetic considerations of a perfect gas we have to 
make use of. the experimental law, viz., pV — RT. Hence 

RT = jMC 2 , x 

.... (17) 



fV 



M 



Thus C a is proportional to the absolute temperature which may 
thus be considered proportional to the mean kinetic energy of transla- 
tion of the molecules. This is the kinetic interpretation of tempera- 
ture. Hence, according to the kinetic theory, the absolute zero of 
temperature is the temperature at which the molecules are devoid 
of all motion. This deduction is, however, not quite justified since 
the perfect gas state does not hold down to the absolute zero. The 
interpretation given by thermodynamics is somewhat different 
(Chap. X) and is more reasonable. That does not necessarily require 
that all motion should cease at the absolute zero. 

i put Mz—Nm where m is the mass of a single molecule 
the iiuniljer oL molecules in a gram -molecule, which is usually 
called th> 

□ put R/N = k where k is a constant, k is known as 
ttoltzrnann's constant. Hence we get 

p= (N/V)hT = nkT f 

win: notes the number of molecules per c.c. Further from (17) 

fcJViiw" - $MT t 

or W^PT", . (17a> 

Le. t the mean kinetic energy of translation of one molecule is §&T* 

Exercise. — Calculate the molecular kinetic energy of 1 gram of 
heli: : : N.T.P. What will be the energy at I00°C? 
From (17) the kinetic energy is equal' to 

Energy at 100°C = 8.5 X 10° X 373/273 ™ 1.16 )< 10*° ergs, 
15. Distribution of Velocities. Maxwell's Law. — Tn the above we 
were concerned only with the mean square velocity and did not 
care to find the velocity of every molecule. But for studying the 
properties of the gas further we must know the dynamical state of 
the whole system. It Is easy to see that all the molecules cannot 
have the same speed for even if at any instant all the molecules 



- 



possess the same speed, collisions at the next moment will augment 
the velocity of some and diminish that of the others. As the number 
of molecules is very large (2.7 X !0 ia per c.c. at N.T.P.) and they are 
too small to be visible even in the ultra microscope, we do not interest 
ourselves in the behaviour of individual molecules. We treat the 
Lilem statistically and apply the theory of probability. We shall 
illustrate this by means of an example,' In a big city there are 
persons of alleges and we find the number of persons whose ages 
lie between definite ranges, say between 10 and 15, 15 and 20, and 
so on. So in an assemblage of molecules where the molecules have 
all velocities lying between and infinity we find the number of 
molecules dn t possessing velocities lying between c and c -f- dc. In 
the steady state this number remains constant and is not modified by 
collisions. This number is given by die distribution law of Maxwell 
which states : 

dn c = 4iraaV**Vi: > .... (18) 

ere n h the number of molecules per cc, and a = \/bfir = 

mf%vrkT. But we cannot say what the velocity of any individual 

molecule selected at random is. We can only say that the probability 

that its velocity lies between c and c -f- dc is ' 

Thus the distribution law gives a complete knowledge of the gas only 
in a statistical sense. 

A slight transformation ^putting &c s — x 2 ) will show that the 

number dn = 4wjr~&x*e~^d% t which helps us to represent the law 

iphically,. Let us plot the function y — £tt~* *""**x 2 against x 



ti Curve illustrating;: it 



Maxwell's distribution law 




Fig. S. — Curve illustrating Maxwell's Law. 
(Fig, 5) . Then the number dn of molecules whose speed lies between 



7b 



KiNirilO THEORY OF MATTER 



[CHAP* 



x and x 4- dx is proportional to the shaded area. The ordinate y gi 
the fraction of the number of molecules posse- correspond- 

ing to x, and £ro.m the curve it is obvious that the probability con 
pon< :; — 1 is greatest, while it is considerably less tor % — 2 or 

x — \. "Hence we can approximately treaty die whole gas as endov. i 
with die most probable velocity corresponding to x — 1. 

16. Average Velocities.— We must distinguish between two- 
velocities, the square root C of the mean square velocity, and the. 
mean velocity c. The former is such that its square is the average 
of the squares of the velocities of the molecules. Thus 

3 UT 



-■*-±l 



which we have already obtained in 
The mean velogit) 



(17). 



n 



dn,.=- 






tm 



(20) 



(21) 



The most probable velocity a is that value of c for which N e the 

number of molecules with velocity c is maximum. Hence for such 

dN 

= 0. TMs relation gives «■ Substituting 






get 



v™=v\ 



c. 






An consequence of the large molecular velocities is 

seen in the almost complete absence of an atmosphere from the 
surface of the moon. Dynamical investigations show that if a particle 

is projected from the Earth with a velocity exceeding \/2gr where 
g is the gravity at the surface of the earth' and r its radius, it wfl 
never return to the earth and will be lost in space. This critical 
velocity Is about II Kilometres per second for Earth and 2,1 kilo- 
metres for the moon. Russian scientists have recently (2nd 
January, 1959) been able to launch a cosmic rocket "Mechta" which 
overcame the Earth's gravitational barrier and Hew past the moon 
space to become the first artificial planet of the sun with an 
orbital period of 447 days, 

_ Calculations show that the average velocity of hydrogen at v 
ordinary temperatures is about 1.8 kilometres per second and accord- 
ing to Maxwell's law a large number of molecules have velocities much 
greater and also much less than this. Thus all molecules having 
than the critical will escape from the planet. Due- 
certain fraction will always have velocity 



to molecular collisions 



greater than the critical and will escape. This loss of the planetary 
atmosphere will continue indefinitely. It is for this reason that there 







LAW OF EQUtPARITXON OF ENERGY VI' 

tically no atmosphere on the surface of the moon while the 
atmosphere of the Mars is much rarer than that of the earth. 

17. Law of Eqnipartition ef Kinetic Energy.— We next proceed 
ileal with the law of equi partition of energy. It is hetter to> 

introduce here the idea of degrees" of freedom of a system. Suppose 

ch an ant constrained to move along a straight line ; it has. 

then only one degree of freedom and its energy is given by %mx 3 . If 

, allowecrto move in a plane the energy is given by -hmx 2 4- hny*. 
An ant cannot have more than two degrees of freedom, but a bee 

ch is capable of flying has three degrees of freedom, all 
translation. Thus a material particle, supposed to be a point, can 
have at most three degrees of freedom. A rigid body can, however, 
not only move hut also rotate about any axis passing through itself, 
The most general kind of rotatory motion can be resolved analytical^ 
into rotations of the body about any three mutually perpendicular a: 
through a point fixed in itself. Hence the degrees of freedom contri- 
buted by rotational motion are three. We may now state the defini- 
tion of- the term 'degrees of freedom'. The total number of inde- 
cent quantities which must be known Before the position and 
figuration of any dynamical system can be fully known is called 
the number :r , : 

Now it can be shown from rigorous dynamical considerations that 
nergy corresponding to every degree of freedom is the same- 
as for any other, i.e., the energy « equally distributed between the 
various degrees of freedom. This is the law of cquipartition of 
kinetic energy and was arrived at by Maxwell* in 1859. Boliamannf 
ended it to the energy of rotation and vibration also. It can 
further be shown that the energy corresponding to each decree eJ 
freedom per molecule is XhT%. This law is very general, but we V 
not attempt here to prove it. 

Thus if any dynamical system has n decrees of freedom 
e^fy I tt at T*K is n )< hTtT, ' 

18. Molecular and Atomic Energy,— The above theorem is 
useful in calculating molecular energy of substances. Let. us calculate 
Luc specific heat of -uses. In a mm gas the molecules are 
identical with atoms and if, as a Erst approximation, we assume the 
atom to be structureless point, then from the previous consideration s. 
each molecule has got three degrees of freedom and will have tlv» 
kinetic energy equal to &X£hT. In the state of perfect gas the 

* Maxwell, Collected Works, Vol. 1, p. 378. 

fLudwig Boltzmaan (1844-1906). Born aud educate! in Vienna, he wi 

! esaor of Theoretical Physics at Vienna, G raZ| Munich and Leipzie On 

account ot his ^ fundamental researches he is regarded as one of the founders 
of the kinetic theory of gases, 

mean kinetic energy of 



$ Equation (17a) gives ±mc a = $kT 1 Le. t the 



translation per molecule is IkT, If wc assume that the C n< 

distributed between the three degrees of freedom, the energy associate* with one 

,' : °\ t'-.'^om Per mojecule heconins UT. For a formal proof of the 
see the Authdhs' Treatise oh Heat, §3.26 



Ni: 



KINETIC THEORY OF MATTER 



CHAP. 



molecules possess only kinetic energy and no potential energy. The 
total energy E associated with a gram-molecule is N times the above 



expression. It is thus equal to %NkT =z$RT. 
heat at const-ant volume is therefore 

dE 



The molar specific 



C v =\ T - =^\ R = 2 " 98 &d. /degree. 
For all perfect gases we have established the relation 



C,~C t 



- 11. 



Therefore, for a monatoudc gas 

and the ratio of die two specific heats 



J*JL+ Rs!a *±R = 4-96 cal./dcgrce. 



(23) 



y =s 5/3 = 1^& 

These theoretical conclusions agree with experimental results (see 

i able 5, Chap. II) for the nionatomic gases like argon, helium, etc 

The specific heat off polyatomic gases can also be obtained by 

using the equipartition law. The molecule of a diatomic gas may 

be pictured as a system of two atoms (assumed to be points) joined 

"dlv to one another like a dumb-bell. The sysi 

"iti'on to the three components of the velocity of translation of the 

common centre of gravity, two components of the velocity of 

out two axes perpendicular to the line of centres of the 

iras. Thus the system has five degrees of freedom and the 

energy E = $RT. Hence 

=$R, G,-1R, y=14. . , . (24) 

This is approximately the case for hydrogen, nitrogen, etc. 

(p. 02'). At low temperai • however, C t falls to $R, as Eucken's 

•riments with hydrogen show, indicating that the rotation hat. 

disappeared. For chlorine C, is greater than -,R. This shows that 

Lhe two atoms are not rigidly fixed bur. ran vibrate in a restricted 

: nner along the line of centres. 

In a triatomk gas a molecule possesses three translational and 

•hxee rotational degrees of freedom and hence 

C „ = fi >< W = SK, C P =4R, 7 =\M,, . (25) 

more complex molecules y approaches unity but is always 

greater than it. Tt is not possible to calculate in a simple way the 

ergy of internal vibration of such molecules since the vibrations 

are not freely and fully developed. 

An expression for the specific heat of solid i may also be obtained 
from the kinetic theory. We can couse ecules of a solid as 

elastic spheres held in position by Lhe attraction of other molecules 
and capable of vibrating- in a simple harmonic manner about a mean 
position. The molecule will have three components of velocity, and 
nee three degrees of freedom. The kinetic energy associated with 
each degree of freedom is ^kT. On the average die harmonic vibration 



i i 



MEAN FREE PATH PHENOMENA 



81 



'•, . ergies and hence the total 
issociated with each degree of freedom is kT. T re, the 

Eor the three degrees of freedom is $RT f 
n ! i heat ia 3i£ = .5.955. This yields Dulong and 

- . I . 
But the kinetic theory of specific heat is unable to explain the 
mi o specific heat with temperature (p. 45), particularly, the 
markctr decrease at extremely low temperatures. Further, the 
jradual and cannot be explained by the disappearance of 
se oT freedom which would involve discontinuous changi 
iplcs of ^R. We cannot assume fractional degrees of fvc 
the principles of classical dynamics and equipartition law fail 
lately. The quantum theory of specific heat has been developed 
whu hi existing facts sai tsfactorily. 

MEAN FREE PATH PHENOMENA 

19. Need for the Assumption thdt Molecules have got a finite 
Diameter.' — We have seen in the previous secti ms that the molecules 

i i ;i ■'. iving at ordin . iperatures with very large velocities ; 

I the case of air it amounts to i ; ; metres per sec. There 

is no force to restrain the motion of the molecules. Hence the 

was raised that the assumption of such large rectilinear 

ies was incompatible with many facts of observation. If the 

particles in ing with such enormous velocities., the gaseous n 

:ined in a vessel would disappear in no time. But we ar 

e that the top of a cloud of smoke- holds together for hours., hence 

: be .some factor which prevents the free escape of particles. 

A very simple explanation was offered by Clan sins. He sftoi id 

that the difficulty disap we ascribe to the molecules a Finite 

ugh very small volume. Then as :■ i irtirle moves forward, it is 

to collide witbl another particle after a short interval, ami its 

tty and direction of motion will be completely changed. The 

path traversed between two successive collisions will he a straight, line 

ibed with a constant velocity, since the molecules exert no fori 
ivei one another except during collision. Hence the path of a single 
cle will consisi of a series of short zig-zag paths as illustrated : 'i 
6. 

.Some of these paths will be long, others will be short. We 

< an define a mean free path A . Add 

I the lengths of a large number 

paths and divide it by the total 

number; this will give ■•' - This 

quantity is of peat importance in 

tudying a class of phenomena, ca1 

port phenomena,, such as vis- 
cosity, conduction of heat and 
ion. 

20. Calculation of the Mean Free Path* — We shall give a very 
n :• method or calculating d i mi an free path approximately. We 

6 




i- 5— Illustration of h 






kinetic: theory of matter 



[ CHAP. 

the one 



m 

and 



.,!:, the simplifying assumption that all molecules except the one 

under consideration arc at rest. The moving molecule will o 
with all those molecules whose centres lie within a distance a train 
its centre U being the diameter of the molecules), and are thus 
contained in a sphere of radius described about tne centre of the 
moving- molecule. As the molecule traverses the gas with velocity 
it wdl collide with all the molecules lying in the region traversed b> 
its sphere of influence, The space thus traversed in a second is a 
cylinder ot base W 2 and height v, and hence of volume it<**v. .t the 
number of molecules per ex. is », this cylinder will enclose ttc-l-. 
itres of molecules and hence the number v of collisions per second 
■ n ffff 2 vnj The length of the mean Iree path 



V fffl ' 



1 



111 the above we assumed the other molecules to be at rest. 
Maxwell corrected this expression by introducing into the foregoing 

, itions th( motion of all molecules according to Maxwell 
lisi itioi and obtained the result 

A— J— W) 

■:TTCT~ 

lumber o isions suffered by a molecule pel 

21. Transport Phenomena.— 1 be distribution law expressed b) 
o be put in the [otto 

dw, . . : 

where dn is the number of molecules with velocity romponenl 

du, v and v -f- do, w and zv -|- dm respectively. 11 
nth mass motion represented- by 

if'-' F*+W*) 

■^> du dv <m\ 



dn = n{, L 






,,. I ici-e U = u - U&J v ' '* and W = « - B 

11 the gas is not in a steady state any one of the following cases, 
:• or jointly, may occur. (1) Firstly, u, x , v 9l xti® may not have 
he same value in all parts of the gas 50 that there will be a relative 
motion o£ the layers of die gas with respect to one another. We 
have then the phenomenon of viscosity. (2) Secondly, T may not 
be the same throughout,, then we have the phenomenon of conduc- 
tion, viz., heat will pass from regions of higher T to regions of lower 
T. (3) Thirdly, if n is not the same everywhere we have the case 
of diffusion, i.e.', molecules diffuse from regions of higher concentra- 
te regions of lower concentration. It is thus obvious that 
viscosity, conduction and diffusion represent, respectively the transport 
of momentum, energy and mass. These are called transport pheno- 



VISCOSITY 



83 



ina. These phenomena are brought about by the thermal aeitation 
the molecules. But the molecules move 'with very lame velo- 
cities, wink these processes are comparatively very slow. The 
cause of this anomaly lies in the frequent molecular collisions Hence 
■ iiiv of these phenomena is most conveniently done through the 
chmum of mean free path. The molecules carry with then 
tarn associated magnitudes and thereby tend to establish equili- 

22. Viscosity.-- We shall first discuss the phenomenon of vis- 
it)'. Here we shall give an elementary treatment of the pheno- 

nenon based on considerations of menu free path. Consider a gas 
-i :>n and choose a horizontal plane xy such that there is a mass 
motion , ot the gas parallel to xv-plane but no mass motion aTong 
K *axis. Assume that the mass-velocity „ n increases upwards as z 
ases. The molecules above the plane z = z» possess, on the 
rage, greater momentum than those below it and hence when 
xnles from either side cross the plane there is greater transom* 
tentum downwards since the number of molecules movXsl 
wy is the same, there being no mass motion parallel to the V-axis 
We can consider every molecule, on the average, to traverse a 
equal to the mean free path and then suffer a collision. 
le velocity gradient is du /dz the difference in the mean molecular 
oss two planes separated by a distance* A is Xdujdz The 
1 » oi a molecule being m , the difference in momentum is mUu^fdz. 
m. due to heat, motion, the number of molecules moving along the 
v-ax is must be, on the average, the same as that moving akmg the I 
the ***«. Hence one-tlnrcl of the molecules may be considered 
moving along the .axis both up or down, or onlv olS De 

he oWrvlr m0 T 1Dg 01 ** "P™*" direction. Consider unit area of 
the observation plane z . The number of molecules crossing this area 

Si' thf ] " P r rd diimi ? n ** be f ** s ^»ere n /the m, 
"Jar density and S the mean molecular velocity corresponding to the 
tperature of the gas, since all those molecules contained in^a cylin 
der of base. unity and height * cross the unit area in one second 
Hence the momentum transferred across the plane in the upward 

'ion is lnc[ G -mX^) where G is the momentum correspond- 
to the observation plane. Similarly, for molecules going down- 
the momentum transferred is ^{c+mX^) . Hence the 

il momentum transferred downwards ,, -.,;...:. ftp Tllis -„ 

d z 
L an accelerating force on the lower layers. Or the lower layers 



Vtmily we m us t uke the average reSoIvetl pa rf nf , 
1 '"• ' ■ out to 1 :• instead of A. 

Rigorous calculation shows that this will be fytc. 



along the - ; 



■84 



y OF MATTER 



• 



| retard this fester Layer by a force equal to this. By definition of 
;:siu q this force must equal »jd Hence 

... (30) 

e p is the density of the gas. 

23. Discussion of the Result— We have established thai 



but X ==— — - (p. 
V2 mro* 



82) j hence 



wo* 






Now is proportional to the square mot of the ab 
perature, hence the coefficient of viscosity is independci pres- 

sure or the density of the gas, provided the temperature is co 
This deduction is 'rather surprising and was first regarded with suspi- 
cion. Mayer and Maxwell subsequently showed experimentally that 
lw actually held for prcssm: s lyi *een 760 mm, and 10 mm., 

unci this was a striking success for the Kinetic Theory. Bin 
both limits the law fails. At higher pressures deviation from the law 
ected since the mtermolccular forces ran not be i, At 

•e path A gradually increases till ii becomes 
arable with the dimensions of th ' then 

remains constant Any further decrease of pressure n and 

pressures the coefficient of \ iscosity 
ire. 
Again i) for different gases should vary 

is found rue. 

24. Conduction.— Let us now find an expression for the 

i. In this case th< rature 

ies from Iayi i Lo layer and it h the civ 
which is tran i one lai mother. Considering the 






dUn 



the 



JnfA 



energy gradient " instead of the momentum gradient m -• 
I transfer of energy downward? per unit area becomes 

M_ 

dz 
where E denotes the mean molecular energy pertaining to any 

If K be the conductivity of the gas, -j— the temperature 

client, the flow ol energ) across unit area in the downward direction 

is JK , - where / is the mechanical equivalent oT heat. The 

dT f fE dT 

j K _ = ^ Xjt _ _ 

d'E 1 






' ' an — 






III.] 



\ ,\U I O] '..NTS 



when c is expressed in heat units. Further considerations show 

i above result must be modified into K zeifC v where % is somi 

and the variation of c- is small, the variation 

ol conductn Lth pressure and temperature follows in general the 

tine coi variation of viscosity. Thus conductivity like 

dependent of pressure. This was verified experimentally 

in and others. We shall no* consider the phenomenon ot 

on as ii is somewhat more complicated. 

25. Valae of Constants. — The value of the root mean square 

a gas was calculated numerically for nitrogen on p. 75. 

lean velocitj c can be calculated from (21) and is equal to 

C 4.93 X I« 4 = 4,5 X 10* cm. /sec. Then (30) gives, on substitm- 

166 X 10- 6 gm. cm* 1 sec-', 






k- ?*i« I0 " fi -9xl0-«. 

P c ~ 1-23 :I0-" Mv/IlM ,u cm - 



i collision* suffered by a molecule per second is 

Assuming h = 2,7 x 10" per c.c. we have 

WSTmrAV = ^r':' •! ! y I ,j-e) 5 ! l0_Scm - 

s will give an idea o\ the numerical magnitudes involved 
kinetic theory of gases. We give below several constant'; 
i at a C and atmospheric" | 



/,./,■' ! 



moli 

10* 


1 city 
in 

me! 
per sec. 


Viis 
7X10^ 
\a gm, 
cm "' 
lee" 1 


srmal 
conductivity 
K X 1C 

incaL cm. 1 

■src.-i "C" 1 

318 

52 
56 
339 
38.9 


Mean 

free path 

Ax 10* 

cm, 


Molecu- 
lar dia- 
meter 
10* 
cm. 


C!, 


43.1 

45 2 


493 
461 
1311 
413 
30? 


166 

12 I 


9.44 

9,95 
28.5 
10.0 

4.5" 


J. 47 
3.5 
3.39 
218 

3.36 

4 ■;. , 



taken from Kayc and Laby f T if Physical and 

' . f-ongman. Greet. & Co., 1948. .. at accuracy is claimed 

values given. 



36 



K.JUNE11C THEORY Of MATTER 

Book Recommended. 



[chap. 



A good account of the subject matter of this chapter will be found 
in Eugene Bloch, Kinetic Theory of Gases, English translation 
published by Me time n and Co. 

Other Referen/r . 

1. Jeai tic Theory of Gases (1940), Camb. IT P. 

2. Kennard, Kinetic Theory of Gta 191 raw-Hill. 



1. 1 [AFTER [V 

EQUATIONS OF STATE FOR GASES 

1. Deviation from the Perfect Gas Equation.— By the b 

Quatiorj oJ State" is meant the mathematical formula which 

the relation between pn volume and temperature 

of a substance in any state- of aggregation. If any two of these 

quantiti mown, the third has a fixed value d pending uniquely 

o and can be determined if the equation of state is known, 

ill is is seldom possible. According to the laws of Boyle and 

Charles, we have for a perfect < 

pV = RT J) 

This is the Equation for a perfect gas. 

But even Boyle himself Eound that the law held only under id 
ons, viz., high temperatures and low pressure, while unaei 
ordinary conditions it did not correctly represent the true state ol 
actual gas. Foi every temperature a curve can be 
drawn which has for its abscissa the volume and for its ordinate the 
responding pressure of the enclosed substance. Tlr ■■. are 

led isothermal*, I £ equation (!) wee. isothermals ought 

to i ingular hyperbolas parallel to each other, but experime 

5 ho n Ehe case, The most extended earlier 

.ttions are due to Regnault. He applied pressures up to 30 
heres while the temperature was varied from 0° to 100 G. He 
duct pV as ordinate ag ; - si a abs< Lssa i ■■■ 6, 

7 inira ilii ' I 1 he curves ought to be straight lines parallel 

to the x-axis; actually, however, they were inclined to it. He foui 
that for air, nitrogen and carbon dioxide the product pV dei 
with increasing pressure, while for hydrogen it increases. iU- also 
found tb n abnormal Joule-Thomson effect (Chap. 

VI). Thesi facts led him to describe hydrogen as "more than 
perfect". 1! equation (1) were true the product pV ought, to remain 
constant; thus these permanent gases were shown to be imperfect. 
Later work by Natterer, Andrews and Cailletet in 'in- 

sures confirmed the idea that the actual gases showed consid 
deviations from equation (1). Andrews' experiments are oJ 
fundamental importance as they throw much light on the actual 
iviour of gases and form the basis of an important equation of 
t'irr proposed by van der Waals. Andrews 1 experiments are 
■ibed in the next section. 

The most thoroughgoing and exact experiments are due to Amagat 
investigated the behaviour of various gases up to a pressure ol 
tmospheres. His results particularly" with CO a (Fig, 7) and 
1 1 1 . 1 . • 1 1 • showed that their behaviour is very complicated. 

A different method has been utilized by. K. Onnes who, investi 
i ihe behaviour of several gases at very low temperatures and 
that none of the numerous equations of state proposed correctly 
us the results of experiments. He finds that at any k ■:■.■ 



88 



i J CIONS OF STATE FOR 



[CHAP. 



ANDREWS' JIXPERIM) 



B9 



ture the results are best represented by an empirical equation of 

the type 

pVstA+Bp+Cp + Dp^., . , . -. (2) 
e ^, /J, (7,.., are constants for a fixed temperature, buj vary 
with temperature in a complicated manner. As many as tw« 
constants are used; they are called vinal coefficients. A is simply 
equal to RT while the es of the coefficients uf higher terms 

diminish rapidly. Holborn and Otto, following Onne* J method, 
studied several gases up to 100 atmospheres and between the tempera- 
tures of -183°C and -J- 400*C, and found that they need cake only 
four cons tits. They give the values of these constants for vai 
gases at different temp era Lures. 

The coefficient H •-, •. particular importance. For all gases h 

varies in l simti; way: at low 
temperatures it Has a 
value, then it gradually in- 
reases tnd becomes 

positive. Now if at; am 
peratnre B — ^ and C, D are 
negligible as usual, then 

dp 

Hence, at this -a ture' 

i.j be obeyed up to 
pressures. This tem- 
rnljrd the 

2. Andrews' Experiments. — 
While engaged in ttift attempt 
to liquefy some of die so-called 
permanent gases — an important 
problem of those 
Andrews,* in 1RG9, was led to 
study the isothermals of carbon 
dioxide. His apparatus is indi- 
cated in Fig. 1, aft is a glass 
tube whose upper portion con- 
sists of capillary tube nn.i :'- 
narrower than the lower part. 
Carefully dried carbon dias ide 
passed through the tube for 
several hours arid then the tube 
Fig. 1.— Andrews' apparatus. sealed at both ends. The lower 

end of the tube was immersed under mercury and opened, and some 
of the gas expelled b 1 heat, so that, on cooling, a small column of 
mercury rose in the tube and enclosed the experimental gas. 

♦Thomas Andrews £18 '-l". ::: :. Born in Belfast, he was Professor oi 
Chemistry at Queen's College, Belfast, irom 1845 to 1879. He is remembered 
his work in . with the liquefaction of gases. 







The tube was surrounded by a strong copper tube A fitted with 

brass Hanges at either end, to which brass pieces could be 
attached airtight: with the help of rubber washers. A. screw b 
; through die lower flange. The tube A contained water 
■•■■ n pressures as high as 400 atmospheres could be app 
carbon dioxide enclosed above e. To register t ne a 

similar rair&Wv tube containing; air was placed on the right siue f -m^ 
in a tube A', exactly similar to A, with which it com- 
municated through the tube cd and thus the pressures in both tne 
tub.es were always kepi equal. The pressure in either tube could be 
varied by means of the screws S or S\ The capillary tube a*- 
be surrounded by any suitable constant temperature bath (not shown;. 
3. Discussion of Results- The curves obtained by Andrews are 
shown in l''ig. 2. Let us consider die isothermal corresponding tc 
13. PC. Starting from the 
right we see .(portion AB) m>r — 

that- as we increase the pres- 
sure, the volume diminishes 
laiderably and finally lique- 
i.ll of the gas begins at a 
pressure of about 49 atmos- 
pheres (point B) , As lon| 

u Lctifi ' ' nues the 
pressure " remains consi.au L 
and the volume continually 
diminishes, more mid more 
of the gas being precipitated 
as liquid. This is indicated 
by the nearly horizontal line 
BC. (The slight inclination 
indicating an increase of pres- 
sure towards the end is due 
to the presence of air as im- 
purity) . At C all the gas has 
condensed into liquid and the 
almost vertical rise: of the 
curve indicated by (JD corres- 
i ds to die fact that liquids, 
nly slightly compressible. 

The isothermal corresponding to 21.5°C is of the same gen 
form but the horizontal portion B'C h shorter. In this case the 
icific volume of the vapour when condensation begins aller 

le that of the liquid when condensation has completed 

mi die corresponding volumes lor the previous curve. As 

tperaturc is raised these" changes proceed in the same direction 

as above, till at 3L1~'C the horizontal pari 1, - just disappeared and 




Specific volume in <■£- 
Fig . nnak for Carbon d 



ailed the critical 




two volumes have become identical. This \s 
isothermal for carbon dioxide. Above this temperature, the horizontal 
part is absent from all the isothermals and as we increase the: pressure 



90 



EQUATIONS OF STATE FOR CASES 



CHAP. 



SSdlv - riTi ■ i " formatI ? n eL&lMi<L but ,,u: volume diminishes 
it becomes nearly equal to , ■•!„;„, , ., , ]iqui j at a 

-Ink lower tcnmey alure . This peculiar! i, o\ ffi isothermal alio ■ 
disappears at higher temperatures as is evident from the isothermal 
for 48,1 °C which is much like the isotherm als for air shown separately 
on the right-hand top. 

We thus see that the whole diagram for carbon dioxide is divided 
iiv the critical isothermal into two essentially different regions. 
Above this isothermal nc liquid state is at ail possible even under 
the greatest pressure, while below k there are three separate regions. 
In the region enclosed by the dotted curve BB'PC'C whose highest 
point P, railed the critical point lies on the critical isothermal, both 
lid and gaseous states coexist. To the left of the line PC and 
below the critical isothermal there is the liquid region while to the 
■ '. PB there is the gaseous region. Now if, by means of gradual 
changes, we want to convert gaseous CO.. at 25 C and 60 atm. pres- 
sure {represented by the point R) into liquid CO. at the same tem- 
perature (represented by the point S) without any discontinue 
appearing, i.e., the mass is not to separate into a liquid and a gaseous 
part with a layer between them, we must avoid reaching the inside of 
the dotted curve BB'PC'C. Thus we heat the substance above: S3 i 
and then compress il till the volume he< i to thai of the 

liquid at that temperature. Next coo] >°C and then reduce the 

sure. Thus starting from the point r which undoubtedly repre- 




o/ ike liquik and easeo ,d was 

[merits of Andre, . 

bon dioxide is compressed above 
•i.i f no liquid can make its appearance, however -rem the pressun 
More accurate experiments show tfrtfc this temperature ; -; 
31.0* and not 3T.-1 . The temperature at SI J i is ca led the e 

(T t i for carbon dioxide. We may define critical tern . 
ture as the highest temperature at which a gas can be liquefied 1 
pressure alone. This is why the earlier attempts to liquefy the 
permanent gases failed, though enormous pressures amounting to as 
hi:,! as b,000 atmospheres were sometimes employed. Tli- pressure 
necessary to liquefy gases aL the critical term is called 

critical, pressure (p t ) and the volume which the gas* then occupies 
at the critical temperature and critical pressure is called the 
me (V,) These three quantities are called the critical 
mtt of a gas. A table giving the critical constants for various 
gases, S given on p, 102, It obviously follows that in trying to 
iquej, -. It uMdw to apply pressure alone if the initial , 

v : :i :: ;; .. 'v : ' ■ '*■ "* «< ■>,.•, ■■■, ,;, ;1 , M ,,, lhr crisi „, 




VAN DEE WAALS' EQUATION 



91 



4 Van der Waals' Equation of State.— 1 he equation of state for 
i gases was deduced theoretically fiom the kinetic theory of 
(Chap. Ill) . There we assumed that the molecules have no 
volume and do not. exert forces of attraction on one another These 
assumptions are correct only for the ideal or perfect gas to which the 
actual gases of nature approximate at low pressure and high tempera- 
ture. Rigorously speaking, however, all gas molecules nave Imite size 
whose importance was first pointed out by Clausius. tor our pr. 
purpose we shall treat the molecules as hard elastic spheres. It is 
clear that at very high pressures the total volume of all the molecules 
will not be a negligible fraction of the volume of the gas and further 
even at the highest possible pressure the volume occupied by the 
substance cannot be less than the volume occupied by the molecules 
when they are most closely packed. It follows, therefore, that the 
free volume of the gas to which the Boyle-Charles' law refers is not 
the *ras volume V but is less than V by a factor b where b is related 
the total volume of the gas moleci 
Another simple way of arriving at this result is from considera- 
tions' of collisions. Consider four balls lying separated from one 
another on a line perpendicular to the wall and let the farthest one 
start to move towards the wall with a fixed velocity, if the balls are 
big, die distance to be traversed is less and hence the last, ball will 
strike the wall earlier than when the balls are mere points. Similarly 
for molecules of finite size the number of collisions with the walls 
and hence the pressure will be greater than for point molecules. Thus 
the effect of molecular size is equivalent to a reduction in the total 
volume of the gas by b. 

Hirn, in 1864, pointed out that the molecules must exert forces 
of attraction ov- one another, hence the energy cannoL be wholly 
kinetic, and potential energy due to forces of cohesion must ah 
taken into account. The correction for Forces of cohesion can be very 
simply obtained. These fortes are of the same nature as those which 
give rise to the phenomenon of surface tension in liquids. The mole- 
cules attract one another with a force which varies ly as some 
power of the distance between them. Thus die force will be appre- 
ciable only for small distances and is negligible for larger on: ■•=. 
molecule in the interior is acted on by forces in all direction's and hence 
these will balance ; but this is not so with a molecule on the surface 
or close to it. The components of the forces acting on it resolved 
parallel to the surface wiM balance but not. those in a perpendicular 
direction. There will be a resultant force acting perpendicular to the 
boundary layer and directed inwards. It is obvious that this force on 
■ le molecule will be proportional to the number of attracting 
particles in the fluid, i.e., to the number n of molecules per c.c. The 
force acting' per unit area of the gaseous boundary will be proportional 
to the product of the above force and the number of molecules in that 
area. Hence the cohesive force /?, acting per unit area of the boundary 
layer of the gas is proportional to ?t~. Now n = N/V where N is the 
total number of molecules and V the total volume. Therefore 
/>, oc l/V 2 . This force opposes the outward motion of the molecules 



92 



EQUATIONS OF STATE FOR GASES 



[chap. 

aiifl thus decreases their momentum and hence also the pressure 
generated by their impact. Hence the pressure will be less than that 
calculated previously by the factor a/pa, where a is .some constant. 
In other words, we must replace p in die perfect gas equation b} 

u 
P + p 1,s 'C the external pressure p on the gas is increased by a/V B . 

Applying both these corrections the gas equation becomes 
[*+£)[v-*)-*T. . . . . 

This is van dcr Waals' equation*. Detailed considerations show 

that b is equal to four times the total vol tune of the molecules. Van 

der Waalsf was the flrst to work out a systematic theory, taking into 

account both these factors, viz., the finite size of molecules and the 

ces of cohesion. Van der Waals' equation is found to hold n 

;i deviations from the perfect gas equation. A comparison of 

tins with the experimental results is given in sec. 6. Various other 

equations of state have been proposed. Some of them are more 

accurate than van der Waals 1 equation in certain regions ; nevertheless, 

latter, considering its simplicity, gives in general, the most 

satisfactory first approximation to the behaviour of actual eases. We 

shall discuss this eqj , re detail. 

5. Method of finding the Values of V and V— \ method of 

- of W and <*> ocairring in van der Waals' equation 
1 5 given below : — ' 

rith-al Data,— In the next 
n from theoretical considerations that the 
■e T„ critical p: l( and critical volume 

i obeying van der Wa > , vely givt:y 

• ■ (5) 

■ (6) 




1 c 27!»R ; 






Pi H p c 

Thus j I T * are experimentally determined f a' and 7/ can be 
I wit* the help of equation (6). It may, however" be 
pointed out that the method is not very achate u the \-as do t 
no obey van der Waals' law accurately near the critical pomt. But 
ot of much consequence as V and <b' themselves defend 
upon temperature and volume. The values of V at d ' >' Lffev 
miportanl gases are given in table 1. They refer to 3 rcfof 
:u. X. I .K, and are determined by this method from equ I ^ 

on hJS \t ^'a^'f ° f **"** thls «»«*■ "» ** Author,' Treaty 
tJohatmes Diderik ran der Waals (1837-1923) was born at Lcvdeu m U&t 

1)3 Prize in 1910 ^ *"*"* « ^T**** and was Warded 
< r W)-1S^ : ^t ! ^Jr,^ »*** Y !a the table by 



D S< , ssi ON OF VAN DTTJl WAALS EQUATION 

' ff » 7. — j:\v./r."'.x of 'a' and 7/ /or some gases. 



. 



Substance 


a X W« 

in atm. )< cm. 4 


& x 10* 
in ex. 


ium 


6.8- 


106 


Argon 


2m 


1 X' 


Oxygen 


m 


i 


Nitrogen . | 


I 27:1 


lV:i 


Hydrogen 


48.7 


US 


Carbon dioa 


71V 


• 


Ammonia 


8BS 


1GG 



■nes x cm, 4 = 3 -5 x LO 4 



. , ise.— Using the values of T e = 5.3, p e — 2.HS atm., calcu- 
late 'a' and 'b' for helium tor a gram-molecule, 

27JP 7 ! ■ x:a-3x ]Q 7 ) a xJ5-3)a 
■' - 64 " ,*, ~ 64x2-25x1 '01x10 s 
atm. x cm, 4 

5 3x8 ; 3xl0 7 

h 






,8xl-01xiO e 

Discussion of van der 

equation. — We shall 

van der Waals' 



=24 e.e. 



6. 
Waals' 

now disi 
i-ion 

(/>+;; x -« r - 

This is an equation of the 
.] degree in V, hence it 
ows that for every Kalue 
of p, V must have three 
values. Further, iroin theory 
of equations, either all the 
three values are "real or one 
is real and two imaginary. 

Again writing the equa- 
tion in the form 

RT a , n . 

P = yZTl >yT- * (') 

see that for very large 
values of V, p is small and 
in the limit /;=0 when F=Ga. 
Any in, when V is very small 
approaching b, p tends to in- 
finity. Hence the curve must 
have a concavity upwards. 
Further V cannot be less than 



g 

X ■;,,, 



i 



Theoretical curres 
for carbon divxidz 




Specific volume 

Fig. 3— Theoretical curves for CO= accord- 
ing to van dcr Waals' Equation;, 






* Taken from Landolt and BurDSlcin, Pkysikaliscfe-Chemische TabcHcv. 



94 



EQUATIONS OF SI ATE FOR GASKS' 




ns. 



— -.*.«— ««g equation 

by means of graphs lor every temperature (p = ordinate, V = abscissa) 
curves of die type shown in Fig, 3 are obtained. It is readily seen that 
tlie turves resemble, in general, the experimental curves (Fig. 2) 
obtained by Andrews, but if we look for quantitative agreement bv 
trying to make the two sets of curves coincide we are greatly dis- 
appointed. In fact, the agreement is only approximate and qualitative 
party because of errors in the assumed values of V and 'b' and 
partly, because the equation holds only approximately. 

There is, however, a remarkable divergence between the theoj 
iiLiu and the experimental curves in one region. The theoretical 
curves drawn from van der Waals' equation give maxima and minima 
in me region represented by straight lines in Andrews' curves. Expert 
mentally, this u the region where condensation or vaporisation begi 
and die pressure remain, constant as Ion- as the process continues. 

1 his > difference is easily explained when the theoretical curve is. 
properly interpreted. The part bd inside the dotted curve correspo 
to the fact thai the volume should decrease with decrease of pre, 
which a quite contrary to experience. This would be a collapsible 
state, for any decrease of volume is accompanied bv a decrease of 
pressure which tends to further decrease the volume. (This is 
>arent it we imagine the fluid to be confined in a cylinder ') Thus 
state of the fluid inside the dotted curve represents a state of 
and consequently, can never he realised in 
This is why the part bet is not obtained in Andrews' 
ion ab represents supersaturated vapour and is 
med experimentally as, Tor exam hen air con- 

ater vapour is compressed beyond the pofm when condensa- 
iir, without conden^tion occurring. This 
■ from dust or charged ions which ac« 

T-- This state is, however/ m,,ableandteasik 

d by the introduction of particles of dust, etc. Hence the 
portion i M represents supersaturated vapour in unstable equili- 

brium does not occur ,n Andrews' curves which represent oniylZL 
of .table exmihbrium. Similarly, the portion de represents a supS 
hen ted hqmd which is also in unstable equilibrium and is obtained 
experimentally when gas-free liquid is carefully heated Hcmce is 

^nce d re^LT r m An4 ^ ~ Thl ** W-^diver! 

Van der Waals' theory, however, does not tell us when conden- 
sation begins, ,.*., where the straight part commences. A Imp e 
thermodynamic argument* shows that die straight portion shoKe 
trawn i bat the area abca = area cdec, ' l 



mi 




EQUATION 



95 



maxima and minima points on the theoretical curve can be 



I b\ putting .^0. Hence from (7) , by differentiation, 



dp RT 2a _ Q 



< - 



(8) 

(9) 



2a(V-b) s 
RV' A ' 

i ; cubic equation. Hence, for every isothermal there are 

i ir one real and two imaginary [joints of maxima or minima. 

I In- uii vi s below P in Fig. 3 are seen to possess one maxima and one 

n :i point while those above it have none at all. A slight mathe- 

nsiormation will show that the other point of minima lies. 
ion f '<■!.' and hence has no physical meaning. Equation (7) 
mi. I i'.i when combined yield 

.' <V • 
P ~- ~ pa 



;io) 






i ih.; i urve passing' through the maxima and minima points and. 
n by the clotted curve QPR. 

In Fig. 2 all isothernials lower than P cut the dotted curve 

at two points and, therefore, liquefaction can be observed 

nging the volume along them. For the critical isothermal, 

1 1 rraal corresponding to the critical temperature, these 

His have coah nto one. Referring to Fig. 3, it is readily 

iai the isothermal passing through the point P, where P is a 

! in! lej ion for the family, of curves or a maxima point for the 

urve, is the critical isothermal because below P every isother- 

got maxima and minima points while above P there are 

I all. As we have seen above, if the isothermal has g 

i. ; ■ linima points there must be liquefaction of the ; 

i n< I we can find out the position where liquefaction begins. Foi 

e must be two points on the curve having equal 

ure. I he highest 'isothermal for which this condition is satisfied 

tl issitig through die point P since at P the maxima and 

i points have c >a U seed into one. Hence P in Fig. 3 must be 

Identified with the critical point and the isothermal through it with 
tin critical isothermal. Now for p to be the maxima point of the 
1 : have, by differentiating (19) with respect to T r and 

pqu i in- to zero. 

a_ Sa(Vr-Zb) 



i (Ifl'i 



mm! I ' ■mi ■ 



f/4 

V t = 34. 

b — a 
P* • 2"b z 

I 



= 0, 






27m 



96 



EQUATIONS OF STATE FOR GAS] 



[chap, 

r ri? e -^f- C0 T nSta/ be ver y easil V ded «ced from (7) since 

hi the cntrcal isoiherma] ft* point P is a maximum poin't £ \ ' , 

a point of inflexion, and for it both §| and |*£ *? e equal to „,. 
' have therefore 

dp _ _ #7* 2n n 

• •-") 

2) 






•iRIM.F.NlAL STUDY OF EQUATION OF STATE 



97 



=0, 



^bluing (8) and (12) we get p f = 5*, and then with die 



:,nd %£L=* 

, 3 ' 



7. Defect in van der Waals' EquaUon.-fn spite of 
agreement deductions from van der Waal* equation show 

►le deviation from experimental result,. 1 We haVe r? • 

shown that for carbon dioxide the curves drawn I 

SfP. f "■;■'- ^JMkeexp.nmen^ 

fain van der Waals' equation *ivp$ w — a*, ,..t.i 
: II h found tha. p, i, Sr^equ^o 2* W "P""" 8 "- 




I 



Compression 
pamp 



[ he apparatus is shown in Fig. 4. T is the measuring vessel 

lass which ends in a 

M h calibrated capillary tube. 

urt of the vessel is placed 

iteel cylinder. By means of a 

V and a screw E the 

! is held in the steel cylinder. 

mostat surrounding the 

i rt of the tube which 

losed by means of a mantle 

.il T is filled with the gas 

i investigation at atmospheric 

the space between T and 

the inner wall 11 being filled with a 

iiLn ii. in: quantity of mercury over 

ch a quantity of glycerine or 

in oil is poured. Pressure is 

licated from a compression 

1 1 through glycerine to the 

miiig vessel. At high pressures 

tner< ury will rise up to the capillary 

' in and the volume can be easily 

i r from the calibration. When 

iIm pressure is high, it is measured 

by a compression manometer. If the 

is above 300 atm, and is 

ted only from the inside the capil- 

ii l generally smashed, hence Kfc 4.— CajHetct'a apparatus. 
ih' experimental tube should be subjected to pressure from all sides. 
uih ;i pressure tube was built by Amagat. 

Amairat carried out an exhaustive study of 
the behaviour of several gases by the above 
method. In one set of experiments he employed 
pressures up to 450 atm., while in the next series 
pressures as high as 3,000 atmospheres were 
used. 

(b) Apparatus based on the principle of 
x'anable mass, — This method was employed by 
Holborn and Schultze. Hoi born and Otto and 
Kamerlingh Onnes. These investigators worked 
at high pressures and obtained important results. 
Since with increasing pressure the volume be- 
comes smaller, greater error would occur in read- 
ing the volume at high pressures. To avoid it, 
these investigators kept the volume constant and 
used different quantities of the gas whose masses 
were determined. The apparatus becomes some- 
what corn.pl i rated by the presence of devices for 
ntroduction and removal of the gas both inside and outside the 

7 





Fig. S. — The 
1 1 :- sure balance. 



98 



EQUATIONS OF STATE FOR CASES 



[ CHAP* 



experimental vessel- An ingenious pressure balance shown in Fig, 5 
was used to measure such high pressures. The metal block B was 
firmly damped in position and carried the tube T ^frich was con- 
nected to the apparatus containing the experimental gas. The block 
B has also a cylindrical hole in which the cylindrical rod R accurately 
fitted. Between R and the gas in the tube T there was castor oil so 
that gas pressure, transmitted through the oil, tended to raise the 
piston R. This was just prevented by the screw S pressing on the 
top of R with the combined weight, of the frame F and weights W, 
When balance is obtained, i.e., the piston neither rises nor falls, the 
gas pressure p z=mgfa, where m is the mass of R, S, F, and "W, and 
a is the cross-sec Lion of the piston or the cylinder. 

9* Discussion of Results.— Amagat represented his results by 

graphs in which pV de- 
notes the ordinate and p 
the abscissa. The curves 
for hydrogen and nitrogen 
for several temperatures 
hown in Fig. 6. As 
already mentioned, for hy- 
drogen tire product pi' in- 
creases with pressure, but 
for nitrogen it first de- 
creases. The curves are 
straight lines inclined to 
the pressure axis, while if 
Boyle's law were true, 
they would be straight 
lines parallel to the pres- 
sure axis. The curves for 
carbon dioxide (Fig. 7) are typical of all gases. The Isothermals 
50° have a portion of them parallel to the jbF-axis. This 
indicates that the pressure remains constant while the volume varies, 
and corresponds to the condensation of the vapour. Further, it is 
seen that the curvature of the isothermals diminishes as the tempera- 
ture rises. The minima points Cm isothermals gradually recede away 
from the origin, and the dotted curve through them is parabolic. At 
still higher temperatures no minima point is found and carbon dioxide 
behaves like hydrogen. 

This general behaviour of tile isothermals can be easily explained 
from van der Waals' equation. We have 




j.OOO S.OOO 

p In atmos. 



s,ooo 



6 — Amagat's curves (FV again;; 
for different gases. 






m 



In the third and fourth terms which are small, we can make the 
approximate substitution V = RT/p. Eqn. (IS) then yields 




SSION OF RESULTS 



*-*+&-&.-**■ 



99 



(14) 



emperatures the term abp' 2 /K 2 T 2 can be neglected. If we 

then ] us y-coordinate and p as .-^-coordinate, the plot will be a 

inclined to the pressure axis (e.g. curves for H 2 in 







too ia& lg3 
p in atmos. 

Fig. 7. — Curves for carbon dioxide. 

["bus for temperatures above the Boyle point T B = a/bR 
will always be positive as for II 2 in the figure. For tem- 
peratures below the Boyle point as in the case of N 2 and C0 2 in 
ii and 7, the slope will he negative unless the pressure is too 
' • 1 • 1 1 . This can be readily seen from eqn. (14) which holds for the 
al case, since 

slope = £^'- +- J*. . (15) 



ilope is therefore negative at low pressures but becomes positive 

m| ii iently high pressures. The minima point on the Amagat curves 

i<\, the point where the slope changes its sign, can be obtained by 

•qua ting (15) to zero. Thus corresponding to any single value of 

1 are two temperatures given by the relation 

• ■ (16) 



b ~w 



= 0, 



R*T* 

lotted curve through the minima points is approximately 
i' 11 1 in Eqn- (16) shows that the dotted curve will meet the axis 
in she Boyle temperature T B = afbR. Thus Amagat's curves 
In ltd \ ' •:. ilained with the help of van der Waals' equation. 



too 



EQUATIONS OF STATE FOR G 



[QIAy 



10, Experimental Determination of Critical Constants.— We have de- 
fined the critical constants in sec. $. They are constants characteristic 
at every substance, and are oi' fundamental importance as they occur 
in certain equations of state. Their importance in the study of 
liquefaction is discussed in Chapter VI. 

The determination of these critical values is often a task of 
considerable difficulty. Of these Lhe critical temperature is the easiest 
to measure accurately. For ordinary substances* a hard glass tube- 
like that of Andrews and connected to a manometer may be employed. 
Sufficient quantity of the liquid is introduced and the tube surrounded 
by a thermostat which can be maintained at constant temperatures 
ring by very small amounts. The temperatures at which the 
liquid suddenly disappears and reappears are observed, the mean of 
these giving the critical temperature. The critical pressure is the 
pressure at the critical temperature and can be read easily from the 
manometer. The critical volume is much more difficult to measure 
accurately, for even a small variation of temperature by 0-1° C pro- 
duces a large change in volume, and hence the substance has to be 
kept exactly at the critical temperature. The pressure must also be 
exactly equal to the critical pressure since the compressibility of the 
substance in this region is very great. The method adopted was to 
arrange in such a way that a very slight increase of volume low 
the temperature by a small amount and caused the separation of the 
into liquid and vapour, the liquid appearing at the top. This 
initial volume is called the critical volume. The amount of substance 
initially contained in the tube has thus to be adjusted. 

The most accurate method, 
is to make use of the 
Law of Rectilinear Diameters 
or mean densities, disco 
by Cailletet and Mathias. If 
the density of a liquid and of 
its saturated vapour be repre- 
sented by ordi nates and the 
corresponding 1 temperatures by 
abscissae, a curve roughly para- 
bolic in shape 5s obtained (Fig. 
8). In the figure the vapour 
density of nitrogen is plotted 
from the observations of On- 
nes and Crommelin and is 
densities in the two states go on 




Temperature '. itigradeu 
Fig. 8.— Law of Rectilineal" Diameter 



typical of all substances. The 

approaching" each other till they become equal at the critical tempera- 

mctimes a simple apparatus, first suggested by Caffniard do la Tour is 
yea* for the purpose. Il consists of a glass tube shaped like T, w 

arm somewhat broadened and containing the liquid :n question which is 
l air in the larger arm by a column of mercury. The two end 
closed, the air in the longer arm serving as a compression manometer. The 
■" :-.r,-ir:cc of the surface of separation between id and its vapour in 

the sli >rter arm was observed. 




The curve AB is a line passing through the mean of the vapom 
i quid densities and will consequently pass through the critical 

Lempen •. It was first observed by Cailletet and Mathias that for 

this line was straight or very nearly so. The equation 

; y = %(p r \-p n ) = a-[-bt where y is the ordinate and t 

■ in abscissa and p u p v denote the densities in the liquid and the 

lit-; respectively. This law enables us to find the critical 

■ the critical volume, for we determine the densities of the 

..•apoii and liquid as near to the critical temperature 

hen draw the rectilinear diameter. The intersect! 

ivith the ordinate at the critical temperature gives lhe critical 

iy p c or the critical volume. 

For substances like water which attack glass at high tempera- 
la illetet and Colardeau employed the apparatus shown in 
strong steel tube AB, platinized inside to prevent attack, 
hi ins the water "or the substance to be investigated. It is immersed 
im ,i temperature hath LL which is heated by a gas regulated burner, 




l 



Fig. 9. — Cailletet and Colardeaifs apparatus. 

temperature can be kept constant. The tube AB is 

a similar steel tube FG by means of the flexible steel 

Mercury fills part of the lube AB, the spiral CDE 

he tube FG up to the level $ lt above which there is water filling 

tube up to the manometer. Different pressures can be 

i! 'I 'I i I by the force pump as indicated, At S t an insulated platinum 

h is sealed in the side of the wall completes an electric bell 

n the temperature of the bath is raised the pressure of 



hi. I thus its 

ted to 

ipiia] CDE. 






EQUATIONS OF STATE FOR GASES 



[ CHAP. 



2 *0- 



the vapour in OB rises, mercury is forced past Sj in FG and sets 
the bell ringing. Water is forced in by the pump to keep the level 
of mercury constantly at S x and thus Uie volume occupied by the 
water and' water vapour in AE remains constant. The platinum 
vrirc at Sj completes another electric circuit and serves lo sound a 
warning that die whole mercury is about to be expelled out of AB. 

We thus get the vapour pressure curve of the substance. The 
curve is perfectly continuous and characteristic of the substance. 

for water this is shown in Fig. 
10. If* however, we start with 
different quantities of the liquid 
we get the same curve as far as 
M, but above it we get different 
curves. The vapour pressure 
thus appears to branch off at M, 
a point, whose position was 
found by Cailletet to be practi- 
cally independent of the quan- 
tity 'of liquid taken. This tempe- 
rature is the critical temperature 
for the substance, 

11, We give below a table 
of critical constants, taken from 
Landolt and Bernstein's Physi- 
cal -Chemisch e Tab el ten. 




300 320 3-iO 360 3&0 4 00 

Temperature centigrade. 
Fig. 10.- Vapour Pressure 



Table Z— Critical data. 





T c 


P< 


Bi 


RT t 


Gas 


in °C. 


in atm. 


Specific 


Pc V* 






2-25 


volume 




Helium 


-267-9 


15-4 


3-13 


Hydrogen 


-230-9 


12-S 


;v-\ 2 


3-28 


Argon 


-122-9 


48-0 


1-88 


3-43 


Oxygen 


-188-8 


49-7 


2-S2 


3-42 


Nitrogen 


— 147-1 


33 -5 


3-21 


S-42 


Carbon -dioxide 


SI -0 


72-8 


2-17 


3-18 


Ammonia 


132-2 


112-3 


4-24 


412 


Ether 


193-8 


35-6 


3-85 


3 • B 1 


Sulphur dioxide . - 


157-2 


77-6 


1-95 


3 -CO 


Methvl chloride 


148-1 


65-9 


2-71 


3-80 


Water 


374-2 


220 


2-6 





12. Matter near the Critical Point— There has been much dis- 
cussion about the slate of matter near the critical point since the 
time of Andrews. The properties actually observed are: — • (1) the 





MATTER NEAR TUF. CRITICAL POINT 



10! 




and hence there must be mutual diffusion, and the surface tension 
must vanish, i.e., the molecular attraction in the liquid and vapour 
slates must, become equal ; (3) the whole mass presents a very flicker- 

appearance which suggests that there intent be variations of 
densitv inside the mass. This was experimentally observed to be so 
by Hem and others. They suspended spheres of different densities 
inside the fluid when each comes to rest at a horizontal surface 
Saving a density equal to its own; (4) compressibility of vapour at 
rhe critical point is infinite and is very great near that point. As 
pointed out "by Guoy, this explains the variation of density through- 
out the mass observed in (3) , for the superincumbent vapour causes 
the density of lower parts to increase. 

From these considerations the simplest and probably the most 
correct view which was put forward by Andrews appears to be that 
just beyond the critical temperature the whole mass is converted into 
vapour' consisting of a single constituent and should behave like a 
gas near its point of liquefaction. 

According to this theory the critical phenomenon, i.e.., the dis- 
appearance of the boundary between Che liquid and the vapour and 
not its motion, should occur only when the amount of liquid in the 
tube is such that it will fill the whole tube with vapour of critical 
density. If more liquid is present the meniscus should go on rising 
till at the critical temperature the whole tube becomes filled with 

id. If less liquid is present, the meniscus goes on falling till 
at the critical temperature the whole should become filled with vapour 
alone. Experimentally, however, Hein found that the critical pheno- 
menon is observable when the initial density varies from 0-735 to 
1 -269 times the critical density. This is probably due to the property 
(4) as the variation of density inside the mass allows the excess or 
deficit amount to be adjusted. ' The branching of the vapour pressure 
curve at M observed by Cailletet and Colardeau may be explained 
in a similar way. 

Experiments with water by Callendar point to the existence of 
a critical region rather than a critical point He found that the 
ity of the liquid and the vapour did not become equal at the 
temperature at which the meniscus disappeared, but that a difference 
of density was perceptible even beyond that temperature. The criti- 
i 1 1 point is that point at which 'the properties in the two phases 
line equal. 

Books Recommended. 
1. Jeans, Kinetic Theory of Gates, C. U. P. (1940). 
Kennard, Kinetic Theory of Gases, (1938), 
. Glazebrook, A Dictionary of Applied Physics, Vol. I. 



• 



CHAPTER V 

CHANGE OF STATE 

Fusion — Vaporisation — Sublimation 

1. It is a matter of common experience that on the application 
of heat, »ub&tances change their state o£ aggregation. Thus when ice- 
is heated it melts into liquid (water) at 0°C (the melting point) . 



tion point and later the liquid solidifies at the freezing point or 
solidification point. For a pure substance the melting and the freez- 
ing points are identical, as are also the boiling and the liquefaction 
points. 

The temperature at which any change of state takes place is 
generally fixed provided the external pressure is fixed and the subs- 
lance is pure. The fusion point usually varies very little with the 
pressure (it requires a pressure of about 130 a tin. to lower the melting 
point of ice to — 1°C), but the variation of the boiling point with 
pressure is very great. As a matter of fact, it can be easily shown 

water can be made to boil at any temperature up to 0'-"C.* pro- 
vided the sufficiently reduced. Conversely, the boiling 

iderably if the pressure be sufficiently 
These [acts can be readily verified by considering evapora* 
sed space. 

% Evaporation in a Closed Space.— If we fill a glass vessel partly 
with water and evacuate it with a pump, then water will begin 
to boil even at room temperature. If the pump be now cut off, the 
pressure can be measured by a manometer. For a certain definite 
temperature of the liquid, there is always a definite vapour pressure. 
Jf we increase the total space, more liquid will evaporate and fill up 
die extra space. If we reduce the space, .some vapour will condense 
till the remaining vapour exerts the same pressure. If there is a 
third gas., noL reacting with the vapour, the partial pressure of the 
liquid will be approximately equal to the vapour pressure in the 
absence of the third gas. It. is an important experimental task in 
Physics to determine "the vapour pressure of a liquid at different 
temperatures. 

3, Latent Heat. — Black found that the change from one state 
to another is not abrupt, but a large amount of heat must be 
absorbed before the entire mass is converted from one slate to 
another at the same temperature.^ Thus to convert 1 gram of 

* More rigorously, up fcr> the triple point. 

t Wc have already discussed in Chap. II, pp. 33-37, the methods of measur- 
ing quantity of heat h>- Change of State 



»■] 



CftANGE OF PROPERTIES ON MELTING 



L05 



ice at 0°C to 1 gram of 



of water at 0°C. about 80 calories of b 
red The amount of beat required to convert 1 1 gram ot a 
folidlntoTliquid without raising the temperature ,. called the fa* 
ioua ™* , .**"i TW amount of heat is required for overcoming the 

-,, ,, lC mobile enouch to form a liquid. In solids the molecules 
^?ima £ne< Ta? "bralin| about mean equilibrium positions which 
•re fi«d but in liquids they execute rotational ami trans at onal 
motionflnd wanderVou^hout the liquid, Oiough con»derably * 
nered Similarly, to convert one gram of water at 100 C to vapour 
^1 0-C 538-7 calories are required. This heat which is neces- 
sary or pulling the molecules of water so far apart that they beo „ 
qute ndependent of each other (vapour state) r, known as t. 
Stmt heat of vaporisation. It will be -seen that the latent heal varies 
greatly with the temperature of vaporisation. 

A Sublimation,-- Sometimes a solid may pass to a vapour state 

withLSSh^rfirough the intermediate liquid state. Camphor 

rrnrles a good example of this class. On being heated it does not 

;r. but sSnply evaporates. Such a process is died subhmcUon 

and the substance is said to be volatile. 

But we shall see that the process is not peculiar to am parti 
All solid substances possess finite vapour tension 
When this vapour tension 



cular substance. 

at even ordinary temperature 



small we take no notice of it, but with the aid o delicate apparatus, 
ii ran be measured, A substance is said to be volatile only when the 
boiling point at atmospheric pressure is less than the melting point. 
Thus under an atmosphere different from our own, say at the moon. 
even ice which we do not consider volatile would have to be treated 
as such. The moon is supposed to have a very thin atmosphere 
t <1 mm. of mercury), and the temperature is below C. If 
Consider ourselves transported to die moon, our studies will show 
that ice h volatile because on being healed, it will evaporate to the 
gaseous state without passing through the liquid state. It we wain 
water in the moon we must artificially produce a high pressure 
and applv heat to ice under this pressure. Similarly, camphor can 
be melted to a liquid form when it is heated under high pressure, 

5, Amorphous Solids,— Impure substances, mixtures, and non 
■stalling subsLances do not usually have a sharp melting pint; 

in' their case fusion and solidification take place over a short range 
of temperature. This is due to the presence of two or more sub- 
stances which do not solidify at die same temperature^ Examples oi 
hous substances are wax, pitch, glass etc. Glass gradually 
tens throughout its bulk as its temperature is raised and is usually 
irded as a supercooled liquid. 

6, Change of Properties on Melting.— Several properties of sub- 
i ces change in a very marked way when a substance melts. Ous 

Lo the regular arrangement of' the molecules in the solid being 
■ed by the addition of heat. The following are some of these 
fin iperties :— - 



106 



l HANOE OF STATE V 



[ CHAP. 



(1) Change of Volume, — Most substances expand on solidifica- 
tion while a few others contract. To the former class belong ice> 
iron, bismuth, antimony, etc.; paraffin wax and most metals bi 

to the latter class. Good castings can only be made from substances 
of the former class. Often enormous force is exerted by water when 
it freezes into ice. The burs ting of water pipes and of plant cells 
and the splitting of rocks is due lo this cause, 

(2) Change of Vapour Pressure. — The vapour pressure abruptly 
changes at the melting point. The vapour pressure curves of the 
solid and liquid slates are different and there is a sharp discontinuity 
at the melting point (Chap, X, § 41). 

(3) Change of Electrical Resistance* — The electrical resistance 
of metals undergoes a sudden change on melting. When the sub- 
stance contracts on melting the conductivity increases and when it 
expands on melting the conductivity decreases. Table 1* below gives 
the ratio of the resistance of the fluid metal to that of the solid form 
at the melting point for a number of metals. 



Ratio 



Table 1. 
resistance of fluid meted 
rests to n cr o } crystallised metal 



for some 



Substan 


Ratio 


Substance 
Cu 


Ratio 


Al 




1-97 




0-70 


Li 


1-96 




2-0 


Na 


1-34 


Cd 


1-97 


Ag 


T-9R 


Cs 


->:;. 


f'.i 


0- ■!:■ 


Ga 


0-58 


Zn 


2-00 


An 


2-28 


Sn 


2 - 


K 


1-32 







(4) It is also found that molten metals show a discontinuity 
in their dissolving power at the melting point, 

7. Determination of the Melting Point. — The melting point under 
normal conditions can be determined with very simple apparatus. 
The substance may be heated In a crucible electrically or otherwise. 
For high temperatures, the crucible must be of graphite or some other 
suitable material, and the substance heated in a non-oxidizing atmos- 
phere. If the substance is rare, it can be employed In the form 
of a wire (wire-method) . As thermometer, the secondary standards 
are very convenient to use, the thermo-couple or the resistance 

* Taken from Handbwh for Fhysik, Vol, X, p, , : 7. 




DETERMINATION OF THE LATENT HEAT OF FUSION 



107 



thermometer being generally chosen ; such th ermocouples or resistance 

wires must not be thrust directly into 

the melting substance, but should be 

protected by a sheath of protecting 

material say porcelain, hard glass or 

magnesia tubes. Now as long as the 

substance is melting the temperature 

remains constant and hence the E.M.I?. 

of the thermo-couple or the resistance 

the thermometer is also stationary. 

curve is plotted with the E.M.F. 

the resistance as ordinate and time 

abscissa. The horizontal part re- 
presents the freezing of the metal and 
rhp rrvn-psnondintf E-M.F. gives its 



of 
A 

or 
as 



the 




Fig. 



1.— Melting point of 
copper. 



is 



about 10-5 mV, whirl; 



corresponding 
melting point. Such a curve for copper 
is shown in Fig. 1 where SQPMORh, Pt 
couple is used. The constant E.M.F. 
corresponding to the horizontal part 
corresponds to 1084°C, 

8. Determination of the Latent Heat of Fusion.- For determining 
the latent heat of fusion, an ordinary calorimetric method (Chap, H) 

may be employed, e.g.^(l) *e *>«** of ^^ <?. ™ *?"? 
ice calorimeter, (3) the method of cooling, and (4) electrical methods. 
(3) is unimportant, and will not he considered here. 
The method of mixtures.— It is quite simple and has been ex- 
plained in Chap. II. Most of the early determinations of the latent 
heat of ice were made by this method. Thus if M grams of ice at 
0°C arc added to a calorimeter containing water whose total thermal 
capacity is W and initial temperature 0i and if <? a be the final tem- 
perature, the latent heat L is given by the relation 

ATI+Mff^H^-flJ.- .... (0 

An important source of error in the above method lies in the fact 

that some water adheres to the crystals of ice at 0°C. To eliminate 

this ice below C C is frequently chosen which requires a knowledge 

of the specific heat of ice. In this method we are required to find the 

t taken up by ice in being heated. A converse method may also 

be employed, viz., the heat given out by water in solidification may 

found. The most accurate experiments give the value L =zl*d-b 

i ;if. for the latent heat of fusion of ice. 

The method of mixtures has been very conveniently adopted to 

the simultaneous determination of the melting point and the latent 

heat of fusion of metals and their salts. Goodwin and Kalmus ern- 

this method for finding the latent heat of fusion of various 

Salts, A known weight of the substance contained in a sealed plati- 

m vessel is heated' in an electric furnace to a high accurately 

i. limbic temperature. It is then dropped into a calorimeter and 

iiiiLv of heat liberated is determined in the usual manner. The 






108 



CHANGE OF STATE 



[ CHAP- 




2.— Latent Heat oi 
Fusion of Salts, 



electric furnace is specially designed to secure a uniform temperature 
throughout the platinum cylinder which is measured by a Pt-Rh 
thermo-couple. As calorimctric liquid, water was employed below 
450 °G and aniline above that, temperature. 

First a blank experiment gives 
the heat capacity of the platinum 
vessel. The experiment is then 
performed with the substance 
heated to different initial tempera- 
tures extending over a range of 
about 50 °C both above and below 
the melting point, and the final 
temperature of the calorimeter 
noted. From these after correcting 
'* c for the heat capacity of the vessel, 
the quantity of heat Q necessary 
to raise 1 gram of the substance 
from the room temperature to its 
in ial temperature could be calculated. Plotting Q_ as ordinate and 
T the corresponding initial temperature as abscissa curves of the type 
•shown In Fig, 2 are obtained. The discontinuity in the value of Q 
indicated by the vertical line gives the latent heat of fusion, the tem- 
perature at which this discontinuity appears is the melting point T m ,' 
and thi it any temperature gives-'lhe specific heat 

be substance at that temperature. 

This methoi rably improved by Awbcry and F. 

iffiths who have made acci : : termination of the latent heat of 

iveral metals. employed a very special type of calori- 

tei which d that the heated substance could be kept 

surrounded :r inside the calorimeter and the lid of the latter 

rater hud access to it, so that the loss of liquid 

by e iti n ed. 

ctrical Method. — This consists in measuring the amount of 
electrical energy required to heat a mass of the substance below its 
melt nt to a temperature above it. The method was employed 

by Dickinson, Harper and Osborne Cor finding the latent heat of 
i of ice. Electrical energy was supplied to a special calorimeter 

similar to that of Nernst in which ice below D C was placed. Then 



E 



- J n 



c pi dT+L- 



T l n 



Ct,«dT t 



where E denotes the electrical energy supplied per gram. 7*^, T 3 its 
initial and final temperatures and c^ c p ^ the specific beats of ice 
and water respectively. From this relation L can be calculated. 
This mediod is very convenient for finding the latent heat at low 
temperatm 

Methods similar to the foregoing can also be employed for folding 
the heat of transformation of one allotropic modification to another 
•bur we are not concerned with them here. 



v.] 



EFFECT OF PRESSURE- ON MF-LTINC POIN 1 



109 



9. Indirect Method.— Another method of finding the latent heat 
s-ts in making use of the Clausius-Clapeyron relation (Chap, X) 



■Bp 



(3) 



where - the specific volumes of the 'liquid and solid 

respectively, and-^~= the ratio of the change of pressure to the change 

ol the freezing point. In this way L can be easily found. 

10* Empirical Relationships.— It was observed by Richards that 
if ML denotes the latent heat multiplied by die a torn it weight 
melting point, ML/ T m is approximately constant for all 
substances and its value lies between 2 and 3. This generalisation 
is known as Troulon's Rule* But the relation is only approximately 
true, and the value of the constant seems to depend upon the nature 
of the crystalline form in which the substance solidifies. Table 2* 

i how far this generalisation holds. 

Table 2.— Illustration of Trouton's Rule. 







Melting 


ML 




Substance 


Atomic latent 


point 


T m ' 


Crystal system 




heat, in cab 


r m 






Na 


030 


571 


1-7 


) 


K 


570 


m 


1*7 


* Body-centred 


Rh 


520 


512 


1-7 


f cube 


Cs 


590 


300 


1-7 


J 


Cu 


2750 


135ft 


2-03 


j 


Ag 
Au 


2630 
3100 


1234 
1337 


2 IS 

2-82 


I Face-centred 

cube 


Pb 


1170 


'.■■00 


J -W 


Al 


2500 


930 


2-70 


J 


Mg 


1130 


927 


1-22 


} 


Zn 
Cd 


1800 
1500 


cm 

594 


2-60 
2-53 


/ Hexagonal. 




560 


231 


: 2-40 


[J 



II, Effect of Pressure on Melting Point. Regelation— As already 

mentioned the melting point of a substance is not quite fixed; it 

I binges when the external pressure is varied. Equation (3) giving 

»ani!;e in melting point due to pressure has been deduced in 

Chap. X from thermodynamic considerations. This expression clearly 

■ that the melting point of substances which expand on solidifica- 

II; n i. lowered by increase of pressure while the converse is the case 

other class of substances, 

.argcly taken from Hand buck der Experlmentoi-physik, Vol. 8, Part i, 



CHANGE OF STATE 



CHAP- 




Ice belc the first category, it is rfiis of ice which 

accounts for the well-known phenomenon of Regelation, i.r., the 

melting o£ ice under pressure and its resolidification when the pressure 

is released. This property enables us to explain the elegant expert- 

.i of Tyndall - in which a piece of wire loaded at either end with 

weights anil placed on a block of ice finds its way through the latter 

though the latter remains intact. The well-known phenomenon of 

glacier motion is partly due to the same cause. Snow goes on 

positing on a mountain and when the mass attains sufficient height 

the ice at the bottom melts under pressure and begins to flow, bur 

as soon as the pressure is released it resolidifies. The block of ice 

thus continuously shifts down the slope and we have the phenomenon 

• i glacier motion, 

12. Fusion of Alloys. — Alloys, except those having composition 
in the neighbourhood of that of the eutecti* alloy, do not have a 

j. finite melting point, Consider for example an alloy of lead and i in. 
The melting of lead is S27°C and that of tin is 232°G, the 

eutectic alloy having the composition 63^ Sn and &7% Pb. If an 
alloy of a ana 10% Pb is cooled from the molten state it first 

becomes pasty at about 210°C when solidiitcation commences and tin 
begins to separate out, and this continues till a temperature of 1S$°C 
is reached and the remaining liquid mass which has the composition 
08% Sn and 87% l'h solidifies completely. Thus the addition of a 
little lead to tin or a little tin to lead has the effect of lowering the 
inciting point of the pure substance just as the addition of a little 
melting point of ice.' In fan thp behaviour of the 
alloy is just like that of me salt solution depicted in Fig. 1, Chap. VI. 
i with a molten alloy rich in lead i.e., 80% Pb 
the mass first becomes past) at about 275 °C when lead 
parate out and finally the whole mass solidifies at, 183 C. 
-s whenever the alloy is very rich in one component, the molten 
[11 first become pasty on cooling, the paste consisting of crystals 
held in the liquid, and this will be indicated by a halt in the 
cooling curve. On further cooling a second halt is reached when the 
entire mass solidities at 183 4 C< The alloy corresponding to the corn- 
led the eutectic alloy and this tem- 
perature of ]iS;l' c C is called the eutectic temperature. Tf we start with 
the alloy of this composition it will solidify or melt at a definite 
ii nature — -the eutectic temperature. Similarly if we melt an 
alloy of composition other than the eutectic, it will first become pasty 
and then at a higher temperature melt completely. The alloys of 
Other metals in general behave similarly. 

Alloys are of considerable practical importance. Thus ordinary 
soft solder or tinman's solder is an alloy of lead and tin having about 
60% Sn i.& a eutectic mixture of lead and tin. It has a melting 

•John Tyndall (1820—1893), Born in Ireland, ed at Marburg and 

in Berlin. From IS53 onwards he was Professor of Physics at the Royal Insti- 
tution in London, He was well-Jcnown as a brilliant experimenter and was 

i I of ::i::n-i|- 




VAPOR1SATION 

i i than Lhat of tin or lead, and has a sharp and 

nitc |" i .n ure of solidification. An alloy of tin and lead (tin, 

, i i, at 1J)4 6 C; Rose's fusible metal (tin, 1; lead, 1; 

at 94°C; Wood's fusible metal (tin, 1; lead, 2: 
I ; ii LLt ii. 4) melts at G0'5"C though the melting points 
. and tin are 269, 327, 232°C respectively, 

13. Supercooling— It has been stated above that when a liquid 

i: lilies at a definite temperature (its freezing point). 

wever, if slowly cooled in a perfectly clean vessel, 

In down to a temperature much below the normal frcez- 

;iit solidifying. This is known as the phenomenon of 

or surfusion. Water can in this way be cooled down to 

still lower with a little care. Dufour suspended a minute 

ma mixture of chloroform and of sweet almonds which 

• i!i i gravity equal to that of tile water drop and managed 

latter to — 20°C without solidification while a drop of 

mplitlialene could be supercooled to 40 °G (normal melting point 

«l C). 

soling is, however, essentially an unstable phenomenon. 

I I I'oduction of the smallest quantity of the solid in which the 

I pud would freeze at once starts the solidification. Mechanical dis- 

h as shaking the tube., stirring or rubbing the sides with 

■ i.l, or addition of some other solid is often sufficient to start 

Lion. Tf solidification has once started it will continue with 

ii ii nl heat till the normal freezing point is reached. After 

ii Hi'!-' solidification will take place only when heat is lost by 

i libation, etc. Absence of air favours supercooling probably because 

[usi particles contained in it are then absent, 

VAPORISATION 

14. Evaporation, Boiling and Superheating.— A substance can pass 
the liquid state to the vapour state in two ways, viz- (i) evapo- 

i ebullition or boiling. In the former the formation 
es place slowly at the surface at, all temperatures, while 
In the l.ni.'.'i the vapour is Formed in all parts of the liquid a! a 
• i ii tant temperature and escapes in the form of bubbles producing a 
Initially these vapour bubbles arc formed around small 
nibbles of ah clinging to the bottom and sides of the vessel which 
facilitate the process of boiling. If. however, the liquid is carefully 
ii dissolved air and then heated in a clean vessel, its tem- 
i an be raised several decrees above 100°C without its begin- 
ning to boil, but when boiling starts due to disturbances of any kind,, 
i ivith explosive violence, usually called bumping, and the 
temperature falls to 100°C, This superheating is the cause of burnp- 
i prevented by the addition of porous objects. 

15. Saturated and Unsaturated Vapours.- — If the vapour escaping 
1 a liquid either by evaporation or by boiling is collected in a 

it will form an unsaturated vapour. In the case of evapora- 

in'ii in a closed space as in §2, the space will be filled with un- 





112 



CHANCE OI- STATE 



CHAP. 



saturated vapour if all the liquid has evaporated, but if the vapour 
remains in contact with some liquid, it will he saturated. Unsaturated 
vapour approximately obeys Boyle's law but for saturated vapour the 
pressure depends only on' the temperature and not on the volume, 
When saturated vapour is heated it becomes unsaturated or super- 
heated. When saturated vapour is cooled without condensation 
occurring it becomes supersaturated. 

We shall now describe some methods for determining the vapour 
pressure. The range of pressure to be measured varies from 1G -4 mm. 
to 400 atmospheres. In certain eases, pressures as low at I0-* mm. 
have to be measured. It is clear that such wide range of values- 
requires various kinds of apparatus. 

16. Vapour Pressure of Water. — The first accurate determination 
of iTnti pressure of saturated vapour was made by Dal ton. A similar 
but improved apparatus was later employed by Regnault lor finding 

the vapour pressure at temperatures 
lying between 0° and 50°C. Ueg- 
ria nit's experiments were performed 
with the greatest care and extend 
over a wine range of temperatures. 
His apparatus for the range 0°- 50 c 
hown in Fig. 3. Two barometer 
tubes A and B were arranged side 
side, fed from the same cistern of 
M, The space a above the 
mercury level in the tube B is- 
vacuum, while water is gradually 
introduced at the bottom of A till 
il rises through the mercury column 
and evaporates on reaching h. More 
water is introduced till a small layer 
remains floating over the mercury 
surface in A. A constant tempera- 
ture bath DD furnished with stirrer 
(not shown) , and a thermometer 
surrounds a, b f as well as some 
length of the mercury column. The 
dilTerence in the heights of the two 
mercury columns, which ;<• _ observed 
through a glass window with a catheto meter, gives the saturated 
vapour pressure of water at the temperature of the hath. Correction 
must he made for the weight of water in A, for effects of capillarity, 
refraction, etc. 

For re trip: ratines below C C "Rcgnault modified his apparatus to 

ol Gav l.ussac. The top of the tube A was bent round and 

• a spherical bulb which contained water or ire and "was 

surrounded by a suitable bath. For temperatures not much above 

50 C 'C the apparatus already described (Fig. §) could bt: used when a 




Fig. 3.— Regnauli • 






— " 



;al methods 



113 



few 



C 



i; 



loijfl ith would necessary, but Regnault preferred the boiling 

tec. 18) . 

17. General Methods.— The methods used for measuring saturat- 

i sure can be broadly divided into two classes • 

in which the temperature is kept fixed and the 

is determined either manomelxically or by measuring tire 

1 s iturated vapour. This is called the direct or static method. 

: Metliods in which the pressure is given and the temperature 

ii It die liquid begins to boil is determined. This is the dynamic 

18. Statical Methods. — Regnault's method is illustrative of class 
i 1 ) The same method can be adopted For finding the vapour pressure 
ol any liquid provided it ot react with mercury and the vapour 
pressure is neither too high nor too low. A number of investigate 

employed this method. Their apparatuses differ only in un- 
iitiai details. A general scheme of apparatus utilising this 
i md is shown in Fig. 4. A is a small glass sphere, a 
- i lacity, to which is attached a glass tube 
incl another glass tube D with a smaller bore. 
I In; is connected to a bigger globe G and a 
ii' n ury manometer M about 90 cm. long. The 
whole apparatus is first evacuated through the 
itop cock Sj. and then the latter is closed.' Next 
the gas under investigation is introduced 
the stop-cock S 2 and condensed in A by 
cooling the latter; finally S 2 is also 
The sphere A is then surrounded by 
i] in me baths and the vapour pressure col- 
liding to the temperature of the bath is 
a ted by the manometer M. B is a baro- 
to indicate the atmospheric pressure. The 
apparatus is convenient for measuring vapour 
tires from a few cm. to the atmospheric 
For higher pressures a compressed air 
ii 'meter may be employed when the whole 
as to be made of steel, 
\ similar apparatus was employed by Hen- 
I Stock for finding the vapour press > 
ol a number of gases between -f-10 and 
On the same principle Siemens has 
I I vapour pressure thermometry at low 
1 ' For high pressures we may 

ii the classical experiments of Cailletet 
"il' n (p. 101) with water, Andrews' 

Apparatus (p. 88) may also be used, lluiborn 

ind 13a ami determined the vapour pressure 

"' watei ahovi 200 a C by the statical method, 
[he methi also been employed by Smith and Menzies though 

i] i is is ranch different. 



.A 




Fig:. 4.— Determination 

of vapour pressure by 

statical method. 



1 



114 



CHANGE OF STATIC 



CHAP. 






Ill the experiment mentioned above the pressure was measured 
manorjaetrically, but Uie pressure may also be found by determining 
the density of saturated vapour, for assuming that the perfect, gas 
equation holds, we have 

pST 



P^' 



M 



<*) 



Thus knowing M, p we can calculate p. A very simple apparatus 
based on this method was employed by Isnaidi and Gans. A portion 
of the vapour was isolated and its density determined. 

19, Dynamical or Boiling Point Method,— This method is based 
on the fact that when a liquid boils, its vapour pressure equals the 
external pressure on the surface of the liquid, A definite exter- 
nal pressure is applied on the liquid surface by means of a pump 
and then the liquid is heated. The liquid will 'boil at the tempi 
ture at which its vapour pressure equals tile external pressure, 
that, set up by the pump. Hence the vapour pressure corresponding 
to the temperature of ebullition is the pressure exerted by the pump 
and can be read on a manometer. 




Fig. S» — Iiegnault's Vapour Pressure Apparatus (Dynamical method.) 

Regnault employed this method for finding the vapour pressure 
of water betwe 50°C and 200 D C His apparatus is indicated in 
Fig. ft. The copper boiler A is partly rilled with the experimental 
liquid and contains four thermometers be immersed to different depths, 
inside the vapour and the liquid. The upper part of the boiler is 



BOILING POINT METHOD 



115 



cted by means of H to a pump, tfie pressure being indicated bv 

n i cury manometer NM. The reservoir is kept immersed in a 

bath VI and serves to transmit the pressure from the boiler to 

' manometer as well a, to smooth the fluctuations in the pressure 

til ined by the pump. The vapour of the liquid condenses in C 

uul returns to the boiler, thus the same quantity of liquid is used 

over and over again. First, a definite pressure is established by the 

pump and the boiler heated. In a short Lime the readings indicated 

bj i lie thermometers be become steady. The manometer indicates 

trie vapour pressure corresponding- to this temperature. 

The apparatus can be adapted [or all pressures. For high pressure 

■.'• ihe parts must be made of copper and the pump must be a force 

SL,^^ is capable of great accuracy/ By this method 

>rn and Henmng have very accurately determined the vapour 

-sure of water between 50 and 200°G ft to 16 atm.) . Ramsay aid 

ug app^ed this method to the measurement of very small vapour 

■■urts. J heir apparatus (Fig. 

6) consists of a wide glass tube 

i to which is connected a reser- 

• R, the latter being connect- 
to an air pump and a mano- 
meter M. Definite pressure is 
'.<i up by the pump and the ex- 

ttal liquid, stored in F, 
llowed to drop on the cotton 

• 1 surrounding the bulb of 
the thermometer. The tube T 

kept surrounded bv a suffi- 
iy hot bath so that the ex- 
perimental liquid aL once evaporates inside T and the thermometer 
soon reaches a steady temperature, the true boil. 
mg point ot the liquid under that pressure. 

Smith and Memies have devised an inireni. 
ous modification of the boiling point method 
Ineir apparatus is indicated in Fig, 7. The sub- 
stance under investigation is kept in the sphere 
A close to the bulb or the thermometer T, both 
"ttng immersed in some liquid contained in the 
^-tube 15. . The test-tube is dosed airtight 
and communicates with a pump and manometer 
t shown) . It is further surrounded by a bath 
whose temperature can be varied. A 'definite 
pressure is first established by the pump and the 
temperature of the bath gradually raised. When 
the vapour pressure of the substance contained 
m A becomes equal to the external pressure on 
the surface of the liquid in the tube, any further 
increase of temperature increases the pressure of 
tire vapour m A which consequently bubble* 




w 



Fig. 6.— Ramsav and 
Young's apparatus. 



JJc=* 




'''" i and 

apparatus. 






- 



116 



CHANGE OF S' 



[ CHAP, 



\ . 



LATENT HEAT OF VAPORIZATION 



117 



through the liquid in the tube. When this just happens the press un 
recoX by the manometer ^ the vapour pressure corresponding 
to the temperature indicated by the thermometer. 

The boiling-point method has also been applied to metals by 
£e .."cadmium was heated electrically in a quartz or por, 
Jeltin tube and the temperature of the vapour was measured by a 
thermo-couple. 

20. Discussion of Result-Experiments show that the saturated 
vapour pressure of every substance increases as the temperature n 
raised. Hence the vapour pressure f> must be a function oi the tem- 
perature viz ft — f IT) where / (T) must be of such form that its % aluc 
E522 Sh "the' Utperature T, Various empirical relation*™ 
between ft and T have been proposed from time bo time. They hold 
for limited ranges and are by no means quite exact, and universal. 

In 1820 Young proposed a very simple formula 



log p = A + y> 






and 



are constants. 



where T is the absolute temperature ana A, 

KirchhoR in l I Rankine in 1866 proposed quite independent^ 



the form li la 



B 



\tygp =A+ r -fClogT. 



m % 



This formula agrees with experimental results very closely and can 
also i„. il considerations. It is shown in 

Chai that 



dp_ 



a) 



e fs h the vapour pressure. L tbe latent heat of vaporization, T 
loint and v* v t , the specific volumes of the substance 
in the liquid and vapour states. Neglecting v x in comparison with v s 
v 2 by R T/Mp from the gas laws we get 

. - • (8) 



lo g p= R \ Y ,-+u 



Tf. 



Assuming £ constant, equation (8) yields Young's formula (5) . 
•::v-£ v. , we assume L to vary linearly with temperature, i.e., 
L = L n — i*Tj, 
equation {9) yields us formula (6) . In order to get the exact, value 
r -l' pressure corresponding to any temperature we must use an accurate 
expression giving the v due of L as a function of T, 

The above holds for the saturated vapour pressure of a pure 
liquid; when mixtures of two liquids are investigated they yield bi- 
ting results, (1) Tn case of liquids which do not at all mix the 
:. ii- pressure of the mixture is equal to the sum of the vapour pres- 
sure of the constituents, e,g*, water and benzene or water and carbon 



(2) Iii case the liquids are partially miscible the vapour 

in e is always less than the sum of the vapour pressures of the con- 

s and may even be less than that of one of them, e,g., water 

.iii.l ei tier, or water and isobutyl alcohol. In this case each constituent 

Little of the other and the vapour pressure of the mixture 

aains constant over a large range of composition of the mixture but 

falls off for very dilute solutions finally attaining the limiting- value for 

the pure constituent. Thus if we distil a weak solution of isobutyl 

alcohol, we find that the solution in the still becomes weaker and 

I finally there is only pure water in the still and pure alcohol 

in i he condenser! For a weak solution of water in alcohol the reverse 

ase. (3) When the two licjuids are wholly miscible, e.g., water 

and methyl alcohol, the pressure is intermediate between those of the 

arate constituents. In this case whatever composition we start with, 

alcohol always passes to the condenser leaving pure water in the still. 

21. Vapour Pressure over Curved Surfaces. — In the foregoing we 
have considered the vapour pressure over a flat surface. The vapour 
pressure over a curved surface is different on account of surface ten- 

i. Evaporation from a spherical drop produces a decrease in 

area and hence also in the surface energy due to surface tension 

and therefore, it will proceed further than in the case of a flat surface, 

i.e., the vapour pressure over a convex surface will be greater than 

that over a fiat surface. Detailed considerations yield the result 

taA.^1 (9) 

re fa, p denote the vapour pressure over flat and curved surfaces 

respectively, S the surface tension* r the radius of curvature of the 

rface (considered positive for concave and negative for convex) 

and p tjie density of the liquid. These considerations have important 

nsequenr.es in the precipitation of rain and in the phenomenon of 

[]i v. Equation (9) shows that if r is small and n* Le. 3 

rface is convex, p may become very large. Hence, if suitable 

nuclei for condensation are absent a high degree of supersaturation 

■ he attained and inspite of it, no drops will be formed, 

LATENT HEAT OF VAPORIZATION 

22. Tn the- measurement of latent heat of vaporization we have 
i" remember two points; first, that k?\ absolute value is relatively 

'y, secondly, that the latent heat is absorbed or evolved in an 

isothermal change of state. The consequence is that its experimental 

deterj lination is very little affected by the usual sources of error which 

isent in all ealorimetxic measurements. The methods can be 

grouped under three broad headings: 

l) Condensation Methods, — Those in which the amount of 
I it ".ii evolved when a certain amount of vapour condenses h measured, 

(B) Evaporation Methods. — Those in which the amount of heat 
requ i vaporize a given mass of the liquid is measured directly. 



"1 



118 



CHANGE OF STATE 



[ CHAT*. 



v-| 



EVAPORATION ME1 I 



119 




The heat is generally added in the form of electrical energy and can 
1 easily determined. 

(G) Indirect Methods. — Those in which the la Lent heat is cal- 
culated with the help of some thermo 
dynamical relationship such as Clan 
GJapeyron relation (equation 7) . From 
the vapour pressure curve the quantity 
dp/dT is determined and hence L 
evaluated from (7). We shall now con- 
sider the first two methods in greater 
detail 

A. Condensation Methods 

23, Berthelofs Apparatus, — -Reg- 

nauit's experiments are rather of histo- 
rical interest We describe below the 
apparatus ol' Berth Hot (Fig- 8) , In this 
apparatus which is wholly of glass the 
liquid is kept in the vessel D and heated 
by the ring burner B, The rest of the 
apparatus is protected by an insula 
mantle M, The evaporated gas passes 
through the tube T into the" spnal S, 
placed within the calorimeter |C. The 
iral S is fitted to T by a conical 
ground pio an be easily remo 1 

The vapour condenses within the its latent heat 

lorimeter which can be easily measured by 
observing the rise of temperature on die thermometer placed inside a 
jacket. The amount of water condensed is obtained by welch- 
ing the spiral S before and after the experiment The heat, measured 
the heat of vaporization plus the heat given by the con- 
I liquid in cooling- from the boiling point to die final tempera- 
ture of the calorimeter. The open end or S is connected to a pump 
to regulate the pressure under which the boiling takes place. 

Errors are likely to arise owing (1) to superheating of the liquid, 
(2) to minute drops of wai ing carried over by the vapour. As 

the: use of a ring burner causes the heating to be sometimes irregular, 
Kahlenberg replaced the ring burner by a metallic spiral placed inside 
the liquid and heated electrically. 

24, Awbery and Griffiths used a slightly .modified apparatus in 
which the usual calorimeter was replaced by a continuous flow calori- 
meter. The apparatus is shown m Fig. 9. The boiling chamber is 
heated electrically by an inner coil. The vapour passes down die 
vertical tube which is surrounded by a jacket of water through which 
a stream of water flows at a constant rate. The temperatures of the 
inflowing and outflowing water are determined by two thermo-couples. 
There is a third thermo-couple at the mouth of the vertical tube which 



L— Berthelot's . 

• LUS. 



\ Ijjpti htasins «fl 
iij?3ic&Gntei>i} 



gives the temperature of the condensed liquid as it leaves the appnra- 

In this experiment the 
. i pour must be produced at 
a stead 1 .' rate and this is 
achieved by the use of elec- 
trical heating. The latent 
at is obtained from the 
i" mula 

M8=m[L-t*(t s —t l )l (10) 
where 9 is the excess of 
temperature of the outflow 
water over the inflow water, 
M is the quantity of water 
flowing per unit time, m is 
the rate at which the liquid 
is being distilled, t 2 is the 
boiling point of the liquid, 
,*-, the temperature of the Ii- 
td as it leaves the appara- 
tus. 

B. Evaporation Methods 

25. This method wa i 
employed by Dieter id. who 
measured the heat required 
to evaporate a given mass of 
water with the help of a 
fiunsen ice-calorimeter. The 
water was contained in the 

tube A "of the ice- calorimeter (Fig. 3, p. 34) and the heat was measured 
by finding the mass of mercury expelled. Griffiths found the electrical 
energy required to vaporize a given mass of water. We shall describe 
the apparatus used by Henning For precision measurement of the heat 
of vaporisation between 30 p and 100°C. 

26. Henning's Experiments. — The apparatus employed by Hen- 
ning- is indicated in Fig. 10. C is a copper Vessel, one litre in capacity, 
in which the liquid is allowed to evaporate. This is surrounded by 
an oil-bath A maintained at a constant temperature. The heating 

! place through the spiral D of constantan wire wound on a 
quadrilateral mica frame. E is a platinum resistance thermometer. 
Tfu- vapour which is evolved passes through the German silver tube 
H downwards through the German silver tube KK to the vessel P in 
which it is condensed and weighed. The end of H is bent down- 
wards so that no liquid drops can be carried. The vapour is first 
! Is P, and when the conditions become steady 

h i op-cock R h turned so that steam is led to the other vessel P. 
tfter a sufficient quantity of steam has been led to P the stop-cock 
turned to the other side. 




Fig. 9.— Awbery and Griffiths* Latent 
Heat Apparatus. 



120 



CHAKG£ OF STAT: 



[ CHAP- 



The quantity of vapour deposited in the second vessel is now 

Lou id b) weighing and the quantity 
of heat ' supplied " is obtained from 
observations of the electrical measur- 
ing- apparatus. For determining; the 
heat of evaporation at lower pres- 
sure, P is connected to a large vessel 
i i about 5 litres capacity which is 
maintained by means of a water 
pump at the required pressure, 

A similar apparatus was em- 
ployed by Henniiig Tor finding the 
heat of vaporization of water up u> 
I80°C when the pressure reaches 
about 10 atmospheres. Fogler and 
Rodebush used this method for deter- 
mining the latent heat of evaporation 
of mercury up to 200 

For determining the Intent heal 
of evaporation of substances like 
nitrogen, hydrogen, helium, etc. r 
which become liquid at very low 
temperatures, the above principle 
has been utilized by #Dana and 

• r . . t_r * Ormcs, Simon and Lance and 

_ieat ° 

others. 

DISCUSSION OF LATENT HE J DATA 

27. Variation of the Latent Heat with Temperature. — Experimenl 
; : ::j latent heat diminishes as the temperature at which 
boiling takes place is raised. This was no v&a by early 

ors who proposed various empirical foruuilae. Of these 
Thiesen's formula appears to 
have been most satisfactory 
and states 

L^L X (*,-!)* - (11) 

where t t is the critical tempe- 
rature and Lj a constant winch 
tfces the value of L at 
t = t & - 1, This is of course 
based on the assumption thai 
latent heat vanishes at the 
critical temperature, Henning 
showed that between 30° and 
100-C the latent heat of vapo- 
rization of water is given by 
the formula 





L= 538.86+0-5994(100-0. (12) 



Fiff, 11,— Variation of Latent Heat 
of CO j with temperature. 



trouton's rule 



121 




._itrgator. 

shows dearly that the latent heat vanishes at the critical tempera. 
e of CO, which is about SPG This result is quite, universal. 
e exact variation of latent heat with temperature for all substances 
iven bv the thermodynamic formula 

where c e) c s denote Che specific heat of the substance in the gaseous 
and liquid states respectively and v it v t , the respective specific 
volumes, 

28. Troirton's Rule*— As in the case of latent hear of fusion, 

we have here also an important generalization known as Trouton's 

Rule which states that the ratio of the molar latent heat arm- 

the boiling point is a constant for most substances; or 

symbolically, 

&L= constant, .... (14) 

where M is the molecular weight and Tt, the boiling point. The 
ue of the constant is about 21. The law does not hold for asso- 
rted vapours. Table 3 shows that tire law holds approximately for 

mc;;: substances, 

Table 3 — Illustration of Trouton's Ride. 



Substance 


Gra m-mol ecul ar 

latent heat in 

en lories 

Ml 


ii Pin- 
point 
T, v 

4-29 
20-4 
77-3 
96-1 
188-1 
859-5 
809-0 
319 
353 

:■:■ 

680-0 

858 

942 

1155 

1180 

1887 


v , r ML 

Value O: ,. 

1 b 
experimentally 


Helium 
Hvdrogen 
Nitrogen 
Oxygen 

ochlonc acid 
Chlorine 

: anc 
Carbon disulphide 
Benzene 
Aniline 
Mercury 

slum 
"Rubidium 
Sodium 
Zir.c 
Lead 


22 
219 

1310 

1030 

3890 

4600 

6100 

6490 

7350 

10000 

14200 

15600 

18700 

29300 

27730 

4f)000 


5-1 
10 

17-3 
18-1 
20-7 

. 
19-75 

20-8 
21-0 
22-6 
18-2 
19-9 
20-2 
23-5 
24-4 



T 



122 



CHANGE OF STATE 



[ CHAP, 



V. 



DENSITY OF SATURATED VAPOUR 



123 



It will he seen, however, that ML/T is really not quite constant 
for all substances, A simple theoretical discussion shows that it can- 
not be so., for the boiling point under atmospheric pressure is purely 
an artificial point and has no physical significance. It varies enor- 
mously with the external pressure, while the latent heat, varies neither 
in the same direction nor to the same extent. In the case of water 
the Trouton quotient at a few temperatures is given in Table ■:. 

Table 4. — Trouton quotient for water at nt tempera? urex. 



Pressure 



4-6 mm. 
760 




latent heat Trouton quotient 



606-5 cal. 

; 



40 

25-9 
16 



From the variation in L and the boiling- point it is evident that 
the Trouton quotient will go on decreasing as the temperature 
becoming zero at the critical temperature where the latent heal 
rushes. Thus the quotient can have any value from. to 40 and 
it appears to be merely an accident dim • t substances boiling 

undi the value is about 20, # 

DETERMINATION OF VAPOUR DENSITY 

29, sity we mean the spec the vapour 

lir or hv ity. The vapour density can be ea 

known weight of vapour 
ai a certain pressure and temperature. Thus if to grams of die 

v c.c. at the pressure p and temperature j", and p rt 
it of .. of the standard substance (air) at 27i' IJ and 
760 mm, then the vapour density is given by 

w T_ 760 

/V 273 'p W 

The density of the unsaturated vapour can be easily and a ecu 
rately found by any <.uiq of the standards methods* viz. ? of Victor 
Meyer, Dumas, lio'fmann and others. But before I860 there was 
no method for directly determining the density of saturated vapours. 
The methods adopted were all indirect in which the density of the 
unsaturated vapour was first determined, and assuming the perfect 
gas laws to hold up to the saturate:..] state, the density of' the saturated 
vapour was determined. This assumption is, however, not quite jus- 
tifiable, hence these methods can never give accurately the density of 
iturated vapour. Still, however, they are frequently' used especially 

*Fufl details of these methods will be found in any text-book on Physical 
Uxemistry. 





dial of Victor Meyer, which is of considerable practical importance 
is consequently described below. 
The apparatus (Fig, 12) consists of a 

cylindrical bulb B with a long narrow stem, 

ttear the top of which there is a side-tube. 

The lower part of the tube is surrounded 
i suitable temperature bath which is 

.' t epl nished by a suitable liquid boil- 
ing at some pressure in A. Air inside the 

tube gets heated and is expelled at the 

top; after a time, however, a steady state 

is attained when no more air escapes. The 

substance whose vapour density is to be 

determined is enclosed in a thin-walled 

stoppered bottle and placed inside the 

tube. By manipulating- s it is allowed to 
: gently in 15. The bottle breaks, the 

liquid vaporizes and thereby displaces 

an equal volume of air which escapes at 

the side-tube and is collected in the tube L 

Knowing the mass of this air, rhe density 

of fl r of the substance is obtained 

the mass of the liquid taken 

by the mass of displaced air. 

rnst has modified the apparatus and 

could thereby measure the vapour density 

of KC1 and NaCl up to 2000°C while Wartenberg found the vapour 
density of several metals up to 200f.PC, Foe a 
description see Arndt Pkyxifialisk-chernische Tech' 
mk. Chap. IX. 

30. Accurate determination of the Density of 
Saturated Vapour* — In 1860 Fairbairn and Tate 

sed an apparatus by means of which they mea- 
sured the density of saturated vapour directly. 
Fig. IB explains the principle of their apparatus, A 

is a spherical glass bulb whose narrow stem dips 
into mercury contained in the outer wider glass tube. 
I r communicates with the metal reservoir B. 
Both A and B contain some water a hove the mercury 
levels, the latter containing a larger quantity than 
the former. All air is expelled from the apparatus 
and then both the vessels are surrounded by a bath 
whose temperature is gradually raised. The levels 
of the mercury in the two vessels remain con. 
that in A being- always higher than in B due to the 
excess of water in B. This is so as long as there i% 
any liquid water in A. But as soon as the liquid 
in A disappears, the level of mercury in A suddenly rises. This is 
because the saturated vapour pressure increases much faster than the 



Fjjt. 12.— Victor Mej 
Vapour Density Apparatus- 




Fig, 1&— Fair- 
bairn and Tate's 
apparatus 



124 



CHANGE OF STATE 



CHAP. 



I 



pressure of unsaturated vapour obeying Boyle's law. The tempera- 
ture at which this sudden rise of mercury column in A appears is 
noted. At this temperature the vessel A becomes filled with saturated 
or whose pressure may be found by means of a gauge connected 
to B. Knowing the mass of water in A and the volume of the 
enclosed space, ihe density of the vapour at the particular tempera- 
ture and pressure can be calculated. 

Somewhat later Perot attempted to isolate a portion of the 
saturated vapour and to weigh it. This could be very conveniently 
done by isolating the vapour by means of a stop-cock and getting 
it absorbed in dry calcium chloride and weighing the latter. K. Qnnes 
employed another simple method. In a graduated vacuous tube 
different masses of the liquid are introduced and the volumes of the 

mr and the liquid observed. Thus if m and m' grams of the 
substaj introduced and the volumes occupied by the gas and 

the liquid are Vi and v» in the first case, and v\', v? in the .second 
case, and p g , p t represent the densities of the gas and die liquid, we 
have 



whence 



•—-'•■;- i , 



«'=*»!>* +P»'W, 



"r.'^- r 'i. "t 



(16) 



Book Recommended. 



azebrook, of Applied Physics* Vol. 1 1 article 









CHAPTER VI 
PRODUCTION OF LOW TEMPERATURES 

I7e low^nP^wVc obtainable. The study of the laws o per 
SrC can, at least theoretically, proceed on the «n- 
£5fe*te SJ degree, below the ^eUing point of rcc ; a, ;1 £. 
i?.he lowest temperature conceivable. This is taken as the zeio 
of tne aSte^mperaUire scale. In this chapter we shall dtscu^ 
die principles and contrivances by which die region from C to 
absolute zero can be reached. 

PRINCIPLES USED IN REFRIGERATION 

2. For reaching low temperatures we have to utilize proc, ; 
bv which a bodv can be deprived of its total heat content. Ihe tol- 
lowing methods' may be employed to achieve this end :— 

(i) Bv adding a salt to ice. 
m Bv boiling a liquid under reduced pressure. 
hit) By the adiabaiic expansion of a gas doing external work. 
(iv) By utilizing the cooling due to Joule-Thomson effect. 
(v) By utilizing the cooling due to Peltier effect, 
(wi) By utilizing the heat of adsorption. 
(vit) By the process of adiabatic demagnetisation, 
A general theory of refrigeration will be given later in Chap, IX; 
here we shall simply discuss the principles and contrivances for 
utilizing them. 

(i) Adding a Salt to Ice 

3 Low temperatures may be attained by adding a salt to 
Ice This is the same process which was employed by Fahrenheit. 
The cause of this lowering of temperature is easily imdei 
Pieces of ice have generally some water adhering to them, and 
if salt be added to this ice, it is dissolved by the water and 
more ice melts. The necessary heat for this process, viz., the neat 
of solution and the latent heat required to melt the ice, n extra. 
from the mixture itself whose temperature consequently falls own. 



126 



PRODUCTION OF LOW TEMPERATURES 



[ CHAP. 



This is the principle of pvezing mixtures. This process, ho 



owever. 




cannot go on indefinitely. Fig. 1 
shows the freezing curves "ob- 
tained with ammonium chloride, 
the ordinate representing tem- 
perature in °G and abscissa V 
concentration of the salt. When 
the salt is added to ice the tem- 
perature of the mixture chan 
as represented by the Hine AB 
1 the eutectic temperature of 
~]5<8°C is reached. Tempera- 
tires lower than this cannot he 
obtained in this way for when 
more salt is added it no long 

into solution. The curve 
AB represents equilibrium bet- 
I'tff. L— iN-tua aad water ■ ■ .-, ; e n solution and ice while CB 

™n i i .. ^ „ , represents equilibrium between 

salt and solution and B denotes 
e eutectic mixture with a Table 1. — Freezing Mixtures. 
ed composition and fixed tem- 
perature. 

In Table 1 the composition 
of the eutectic mixture and the 
corresponding eutectic or c- 
tempera 

wnmoner salts, i 
rally hydrated salts are employed 

! in that case the oorre! 

ing quantity hydrated salt 

Id be obtained by calculation, 

i raperatures repre- 

the lowest temperature that 

is possible to attain with that 

Freezing mixture, 

(if) Boiuwg 4 Liquid Under Reduce Pressure 

4. Lot ntme may also he attained by allowing a lie 

boil under- reduced pressure. When a liquid evaporates it requires 
U for conversion from the liquid to the gaseous state (latent heat 
of vaporisation) Thus one gram of water at 100*C requh 
calories for complete evaporation. If such liquid be forced by so] 
conmvancc to evaporate rapidly and if the liquid be isolated." rh." 

ig may be produced. 

The oldest contrivance for utilising this process is the crvo- 

Wicated in Kg. 2. The bulb B contains water or some 

volattle liquid and the rest of the space is filled with the lap u? 





1 Aulr. 


En:. 


Salt 


! saJtperlOO 


tempera- , 




grams of the 


lure. 




rture. 




; o 4 


19-7 


-^f 


ZnS0 4 


27-^ 


.6-5 


KC1 


19-7 


- T T • 1 


NH 4 CI 




1 5 - S 


iNO s 


41-2 


-17-4 


o 8 


37 


-18-5 


NaCl 




- 21 -2 


: :i a 


21*6 


-33-6 


CaCL 


29 -B 


-- V. 


KOH 


81*5 


-<o5 






VAPOUR COMPRESSION MACHINE 



127 



of the same liquid. When A is immersed in ice the vapour in it con- 
denses and the pressure of the vapour in B becomes so much lowered 
that the liquid 'In B boils. The latent 
heat necessary for this purpose is ex- 
tracted from the rest of the liquid which 
consequently free. 




Xow-a-days this principle is em- 
ployed in a huge number of refrigerat- 
ing machines both for industrial and 
domestic work. Water, however, is not 
a suitable liquid to use for though it has 
a large latent heat of evaporation, the 
vapour pressure at low temperatures Fig. 2.— The Cryophorus, 
is small The liquids commonly employed are ammonia, sulphur 
dioxide, etc. Two types of such machines arc in use : — (1) Vapour 
compression machines, (2) Vapour absorption machines. The vapour 
compression machines are more efficient, particularly for large plants, 
and require less initial cost; consequently, their use is 'more common 
than that of the other. The only essential difference between these 
two types of machines consists in the manner of compressing the 
low pressure vapour. In the former a motor compressor is used while 
in the latter a dilute aqueous solution at ordinary temperatures is 
employed to dissolve the low pressure vapour and the coneentral 
solution is heated in a generator to expel the gas at high pressure. 
We shall now describe these machines in greater detail. 

5. Vapour Compression Machine.— Fig. 3 shows the essential parts 
oE a vapour compression machine. There are three principal parts 



Water ou 



P=1L Etmos js , 




CoMsto 
apftse 



p=2'5 atmoa. 



p~j] Liquid ammonia 

pffil High pressure ammonia 
,ow tiresaure ammonia 



V 



Wig. 3.- Essential parts of a Vapour Compression Macbin 

the compressor P, the condenser C and the refrigerator or evapo- 
rator R. The cylinder of the compressor has two valves, S and D, 






.28 



PRODUCTION OF LOW TEMPERATURES 



[ CHAP, 



rhe former for the suction of the low pressure vapour from the eva- 
porator and the latter lor the discharge o£ the compressed vapour 
to th<- condenser. When the piston p moves upwards the pressure 
in the cylinder falls below the pressure in the evaporator and 
the l™ pressure vapour Is sucked in through S and the suction 
pipe. " Dicing the downward stroke Lire vapour is compressed and 
then delivered to the condenser C through the discharge vake D 
and the discharge pipe. The condenser is cooled by cold 
dilating in the outer chamber. On account oE the low ^temperature 
and high pressure the vapour liquefies in C. This liquid passes 
through the expansion valve or the regulating valve V which is simply 
a throttling valve to reduce the pressure of the liquid from the high 
pressure prevailing in the condenser to the low pressure m the eva- 
porator Due to die low pressure the liquid boils thereby extracting 
its latent heat from the cold storage space surrounding the eva- 
porator. This space is consequently cooled. In some cases the 
evaporator is surrounded by brine water kept, in circulation. The 
brine water thus becomes cooled and is taken elsewhere for reFrige- 
rati i purposes. The low pressure vapour is sucked in by the com- 
pressor and the cycle or operations continues. 

In the diagram anhydrous ammonia is supposed to be uti 
efrigerant. The pressures and temperatures of ammonia In 
of die apparatus are approximately as indicated in the 
. 

i : on machine is shown in Fig. 4 

w hicl Iphur di 13 the refrigerant. Vapour 

ively employed in ice-making. 
t and other foodstuffs and for various other Indus- 
trial purp 

6. Refrigerants.— Various liquids have been used as refrigerants, 
important one ai imoniaj sulphur dioxide, ethyl chloride 

methyl chloride. Of these ammonia is most commonly used in 
refrigerating plants, while sulphur dioxide is employed in many 
household , There are various criteria for .selecting a suitable 

refrigerant : (1) The latent heat of the refrigerant should be large 
so that the minimum amount of liquid may produce the desired 
refrigerating effect. (2) The refrigerant must be a vapour at ordinary 
temperatures and pressures but should be easily liquefied when com-. 
d and cooled. Generally a temperature of about 5°F (some- 
below ice-point) is required in the evaporator coils and about 
(about room temperature) in the condenser coils. (3) The pres- 
1 of the vapour of the refrigerant in the evaporator coil must be 
greater than the atmospheric pressure, so that atmos >' impurities 

may not be sucked inside and later block the valves. With cooling 
at room temperature surrounding the condenser the pressure 
necessary to liquefy the gas in the condenser should not be too large 
otherwise the compressor and the cylinder will have to he made very- 
stout and consequently costly and there will be much leak g 
vapour into the atmosphere/ (4) The specific volume of the vapour 



I 



VI.] 



REPlUOERANTS 



129 



of the i ••:: 1 l should not be large otherwise a very large compressor 

will be necessary, 

. 7 h - :: important pii | me common refrigerants are given 

m rable 2. V, is table it k evident that ammonia is the 

gerant. One pound of ammonia will produce 
the same amount of refrigeration as 8.75 pounds of carbon dioxide 
while the pressure hi the condenser in case of carbon dioxide is about 
ater than in the case of ammonia. In the matter of 
spin j! 1 volume, however* carbon dioxide possesses an advantage. 
Sulphur dioxide requires a less stout compressor and condenser than 

rua but for the same refrigerating effect the compressor ha 
be made large. 

Table 2. — Characteristics of refrigerants* 





Ammonia 


Sulphur 


Carbntl 
' dioxide 


Methyl 
irjde 


Freoti 
CCJiF. 


1. Boiling point in 
°F at a tm. pres- 










sure 


-28.0 


14.0 


-108.4 


-10.6 


-21.7 


2. Latent heat of 












1 \ aporation at 5 
in B.t.u. per 












pound 


565,0 


175.fi 


114.7 


178.5 


69.5 


5. Refrigerating 
effect in B.t.u 












per lb. 


474, 


141.37 


JUS/;! 


148.7 


5 LOT 


-L Vapour pressure 
at 5°F in lb/in s . 


34,27 


11.81 


33-1.4 


20.89 


26.51 



5. Vapour pi esi 

at 86°F in lb/in* 

6. Specific volume 
of vapour in 
e vapor? 

cm ft. per lb. 

7. Horse-power for 

a ifrigei Ei n ] 
200 
B.t.u. per min. 



60,45 10B9.0 



6.421 



0,09 



>3.2 



95.53 



4.529 



1.06 



107.9 



].-1S 



. : : 



* A number ants have been 




180 



PRODUCTION OF LOW TEMPERATURES 



[CHAP. 



This method mav be employed to obtain extremely low tem- 
peratures by using liquid hydrogen and helium. These liquids are 
Sowed to boil under reduced pressure when temperature* lower than 
their normal boiling points are reached, 

7. Electric Refrigerator,— Fig. 4 shows a modern electric refri- 

. l . Y? - ' „t „:«..-■ .nwlin™ iiitniYMi-irailv on r. he vaUOUr 




(Oil 

Liatricl SO^ 
I High pressure S0 3 
I Low pressure S0 3 



Fig. 4-— Frigidaire, 

The refrigerator coil R contains liquid sulphur dioxide which extracts 
he:--!, from the surrounding space and evaporates and the low pressure 
vapour collects at the top. This vapour communicates i wtti i the 
suction pipe S and the crank case K to the motor switch W. When 
eno has collected in die top of R it exerts a large pressure 

-which is transmitted through S and K thereby operating the switch 






VI. 



AMMONIA ABSORPTION MACHINE 



131 



'.,'.'. This starts die motor and the latter works the compressor P, as 
a result of which, the low pressure vapour is sucked in through S to 
the ciunk cs and compressed by the piston and delivered to the 
condense] C. The condenser is cooled by a current, of air forced 
across il b) the fan mounted on the motor M but in some cases the 
cooling is bi ought about by the flywheel itself whose spokes are shaped 
like the blades of a fan. The high pressure sulphur dioxide vapour 
on his cooled liquefies and collects in the reservoir T. From 

here on > l of the high pressure the liquid is Forced up through 

the liquid pipe L and enters R through the needle-valve N. When 
enough liquid, has collected in R the float valve V rises and closes 
the valve N. Thus the machine works only when the gas 

essure in R becomes large and liquid is transferred from the storage 
ik T to R only when the quantity of liquid in R becomes less and 
float valve has sunk so as to open the needle valve N. 

8. Ammonia Absorption Machine. — 4j> already slated the ab- 
sorption machines differ from 
the vapour compression ma- 
chines only in the manner 
of converting the low pres- 
vapour into high pres- 
sure vapour. Ammonia is 
the most suitable refrigerant 
use in absorption ma- 
chines, and water is a very 
suitable absorber. Water at 
F absorbs about one thou- 
i id times its volume of 
ammonia vapour but when 
the aqua ammonia 'solution is 
freely escapes from the solution. 

The wolfing of >an ammonia absorption machine will be easily 

understood from Fig.J5. The generator A contains a strong solution 

> nonia gas in water and Is heated by a burner as shown in the 

or by means of pipes carrying steam. Ammonia gas is expelled 

from the solution and passes into the coils immersed in the con- 
denser 1J through which cold water continuouslv Hows. The; gas 
idenscs there under its own pressure into liquid ammonia. The 
liquid ammonia thus formed passes through a narrow regulating valve 
to the spiral immersed in the refrigerator C, where on account oE 
ili low | i me it evaporates. The valve is adjusted to maintain 
desired difference of pressure on the two sicles. Through the 
refrigerator flows a stream of brine water which becomes cooled by 
the evaporation of ammonia. The cool brine colution may be taken 
to .il i \ place for refrigerating purposes. 

mmonia gas formed in- the coils in C is absorbed by water 

i diluta ammonia solution contained in the absorber D arid thus 

the pressure is kept low. The solution in D hecomes concentrated 

and is transferred by the pump P to the generator at the top. Thus 




I" ... 5.-— Ammonia Absorption Machine. 
heated to 80° F ammonia vapour 



132 



PRODUCTION OF LOW TEMPSEEATUREfi 



P 



the supply of concentrated ammonia solution is kept up. Dilute 
ammonia from tin bottom of the generator may be run to the ab 
sorber and concentrated. Thus tlie cycle is repeated and the action 
is quite continuous. 

The difficulties of this machine are that It h a m\\ efficiency 
and the pressures are widely different in the condenser and the 

orator. The low efficiency h due to the circumstance that the 
heat absorbed by the ammonia in the generator is much larger than 
the heat absorbed by it in the evaporator coils. Further the 
machine has a moving part in the pump needed to transfer die 
liquid to the generator, and is costly. All these difficulties are 
avoided in a clever invention by two Swedish engineers, von Platen 
and Muuters, which is :<3 in the market under the name Elec- 

trolux Refrigerator, hi this Dalton's law of partial pi. is used 

to make the tola tire in the condenser and the evaporator 

equal, maintaining at the same time a difference in partial pressures 
of ammonia in the two chambers ; this is accomplished by using an 
inert gas like hydrogen at a pressure of 9 atmos., the partial pres- 
sure of ammonia being •:! atmos. in die evaporator and the absoi 
and ammonia liquefies in the condenser at the pressure of 12 atmos. 
Concentrated ammonia solution is forced up into the generator by 
heat and not by a pump. 

(iii\ ^diasatic Expansion osf Compressed Gases 

9. Cooling produced by the sudden adiabatic Expansion of com- 
pressed gases. — ■ Idenly allowed to 

considerably on account of the 
k it docs in expanding, vide' § 23, Chap. II v 

////?) 'v- ' ■>'>: The cooling: may he so great 
that the ga.s may eve [£y. 

An example which is easily available in a big town is afforded 
i irbon dioxide. If such a cylinder be sud- 

denly opened and a piece of cloth held before it, the issuing ■ 
deposited in the form of solid CO a (called dry ice commercially) . 
This principle was utilised by Cailletetwho first liquefied oxv 
in 1877, He compressed oxygen to a pressure or 300 atmosphere 

j capillary tube cooled to -2SPC by liquid sulphur dioxide boil 
rider reduced pressure and then suddenly released the pressure. 
A mist, of liquid oxygen was formed which disappeared in a few 
seconds. PicteL compressed oxygen to a pressure of 500 atm. and ec 
it to about — 130°C by liquid carbon dioxide evaporating under re- 
re. Then he suddenly released the pressure. Oxygen En 
the form of a white solid was thereby obtained. Tn 1884 Wrob 
obtained a mist of hydrogen while in 1S93 Olszewski obtained liquid 
tin ient quantities, by cooling 'compressed hydrogen with 

liquid oxygen and then suddenly releasing the pressure. Simon in 
1933 produced appj quantities of liquid helium by sudden adia- 

•jiiic expansion of the compressed gas which had been precooled by 
solid hydrogen evaporating at reduced pressure. 



VL] 



[PACTION OF GASES 



133 



The process is, however, essentially discontinuous, hence for 
commercial purposes it was almost discarded; but a novel wr. 
utilizing the principle has been invented by Claude and Heylandt 
foi I air (see sec. 22) » 

W; '"escribed above two types of refrigerating machines. 

There is a third type also which is sometimes employed. This may be 
called the air compression machine because air is here used as the 
[rigerant. In dlis air is first comprcsssed in a compressor, the heat 
of co ion is then removed by passing the gas through coils kept 

, cold circulating water. This cool compressed air then suffers 
adiabatic expansion in the expansion cylinder and becomes consider- 
ably cooled. This cold air then traverses the cold storage space and 
thu Mated and is again compressed. Thus the cycle continues. 

This is the principle of the Bell-Coleman refrigerator largely used 
for the refrigeration of cold storage chambers in ships. 

(h) COOLING DUE TO PELTIER EFFECT 

10. It is well known that when an electric current (lows in a 
circuit from bismuth to antimony through a junction, this junction is 

ooled. Ellis is known as the Peltier effect and may be utilized in 
tducing cooling. This cooling is rather small though semi-conductor 
thermo-junctions have recently produced much more cooling, and have 
been employed in some refrigerators. 

(*■•) Cooling by Joule-Thomson Expansion 

11. This method, is of considerable importance and will be con- 
sidered in detail later in this chapter. 

(vi) Cooling Due to Desorftion 

I2» Charcoal adsorbs a number of gases which are released on 

lowering the pressure, and when these gases escape, a cooling results 

a manner somewhat analogous to the case of evaporation of liquids. 

is is called the "Desorption method" and was utilised by 

i in. In an experiment charcoal adsorbed helium gas at 5 atm. 

I0°K and subsequent de sorption to 0.1 mm. pressure lowered the 

temperature to 4 P K which is sufficient to liquefy heliu: l. 

LIQUEFACTION OF GASES 

13. Liquefaction by application of pressure and low temperature. — 

which are gaseo^ at ordinary temperatures may be 

i the liquid state if'they are sufficiently cooled, and sirnul- 

- pressure be applied to the mass. When pressure 

the molecules come closer together and if heat motion 

sufficiently small, they may coalesce and form a liquid mass, 

ll the liquid so cooled be allowed further to evaporate rapidly, 

Lem| nines may be obtained. The production, of 

in' I leratures is thus intimately connected with the 

. ■ i : hi of such gases which ordinarily show themselves 

efracl 




134 



PRODUCTION OF LOW TEMPERATURES 



[CHAP. 



compression was 



The earliest scientist to try this effect of combined cooling and 
Faradav* who, as early as 1823, employed the 
apparatus shown in Fig. o 
for liquefying chlorn 
One end of the bent glass 
tube contains dr.: substance 
from which chlorine is 
liberated by heat while the 
other end is immersed in a 
freezing mixture. Gaseous 
chlorine coltecLs in the 
other end and finally lique- 
fies under its own pressure. 
By applying this process 
Faraday and others suc- 
ceeded in liquefying a large 
number of gases but some 
viz., oxygen, nitrogen, hy- 
drogen, carbon monoxide 
and methane baffled al! at- 
tempts at liquefaction, 
to S000 arm. were used. 




Fig. &— Faraday* :itus iot Liqpefeclton 

of Chlorine, 



carro 

in 1!- abject 

And i' 

w 

abov 

to which it is .subjected. 



though sometimes enormous pressures up 
They were, therefore, termed menl gases. 

Discovert of the Critical Point— Ths cause of these failures "be- 
ta Andrews' discovery of the critical temperature 
has already been treated in Chapter IV. 
arty showed the importance of the 
Ti, ■ 1 1 clea rly that, for ev er y subs tance , 

Wchmudlyo'ccurs in the gaseous form, there exists a temperature 
bove which 'it cannot be liqueSed however high mas' be the pressure 
, w h subjected. Hence, in order to liquefy a gas by this 

method it must be pre-cooled below its critical temperature. 

The determination of the critical point (p, 100) is not, how- 
ever, easy. The early workers did not in fact, wait for its deter- 
mination. They cooled the gases by ordinary methods as much as 
they could, add then applied high pressures. 

' 14. The principal methods of liquefying -air and other gases are 
the following:— (1) Pictets cascade method which utilises a series 
of liquids willi successively lower boiling points but the principle is 
the same as explained above ; (2) the Limle and Haropsons methods 
employing the Joule-Thomson effect; (3) the Claude and Heylandt 
methods which utilise the cooling produced when a gas expands doing 



♦Michael Faraday (1791-1867), the "prince of experimenters, was bora of 
htunbto i - to London. At the age of thirteen he became errand boy to a 

•.-Uer but later in 1813 he got employment under Sir Humphrey Davy at 
the Royal Institution where be carried on his scientific work and finally succeeded 
Davy in 1827 as Director of the Royal Institution. His greatest work ls the dis- 
covery of electromagnetic induction in 1831. 






VI.] 



SERIES KIlFRIGERATION 



135 



external work. We shall now consider these methods in greater 
detail. 01 these the first is historically the oldest and theoretsqjdlv 
the most efficient hut is somewhat cumbersome and is very little used 
at present. 

15* The Principle of Cascades or Series Refrigeration. — The method 
was first employed by R. Pictet in 1878. In principle it may be 
described as a number of compression machines in series. Pictel 
employed machines containing sulphur dioxide and carbon dioxide 
and obtained temporarily a jet of liquid oxygen by allowing com- 
pressed oxygen to expand adiabatically. Wroblewski and Olszewski 
at Cracow obtained sufficient quantities of liquid oxygen, nitrogen 
and carbon monoxide by the cascade method and determined their 
properties. Olszewski used ethylene as another intermediary below 
carbon dioxide and could thereby cool these gases below their critical 
temperature. Kamerlingh Onnes later employed die combination 
of methyl chloride and ethylene for liquefying oxygen. The prin- 
ciple of the method is illustrated in Fig. 1. Machine (1) utilizes 
methyl chloride. This has got a critical temperature of M3 C C and 
hence at room temperature it can be easily liquefied by the pressure 
of a few atmospheres only. Water at room temperature flows in the 



liquid 

[lea, a 




Fig. 7. — Illustration of the Method of Cascades. 

ket in (1). Liquid CH 3 C1 falls irrto the jacket in (2) which is 
connected to the suction side of the compression pump. Thus the 
Liquid evaporates under reduced pressure and its temperature falls to 

• hi 90 Q C. The compressor returns compressed CH 3 CI gas through 
the tube inside jacket (1), which is shown straight but is really in 
the form of spirals. 

Inside the jacket (2) is placed the condenser coil through which 
ethylene passes from the compressor or the gas cylinder. It liquefies 
and i!i. . enters the jacket (3). There it evaporates under reduced 



PRODUCTION OK LOW TliMPJERATURES 



[CKAP. 



ire and lowers Che temperaAiwre to about - I60°C. Tbj 
tube insi acket (3) oxygen from a and liq 

-essure. Liqu ollected In Ehe sfe D. 

The lowest temp;. obtained by boilin en under redt 

pressure is -218°C which is higher than tl matures of 

(-228.7), hydrogen (-240*0) and helium, hence the method 
of cascades failed to liquefy these three gases. 

The method of cascades is very useful for laboratory purpc 
The use of the compressor can be entirely dispensed with by the use 
uable liquids boiling under atmospheric or red essure. 

: suitable liquids can be selected from Table . res the 

normal boiling point, the critical temperature and the tri int Cor 

the common gases. The interval between die critical temperature 
and the triple poirj icnts the range in which the liquid is avail- 

able. Still lower temperatures ran be obtained by further reducing 
the press. ■ the liquid when it solidifies. Thus, using solid 

nitrogen and a good pump a temperature of about -2 an be 

obtaim :] -above the critical temperature qf neon and it 

is not possible to bridge the gaps a nitrogen and neon and 

between hydrogen and' helium in this way. 
■ 
Table 2— G '", normal boiling point 

and tri ases 



Sul 


B. P, at 1 atm. 


Critical tem- 


Triple point 




pres s 


per:: . 




Mechvl chloride 


- 24.09°C 


143.3 


- 10S.9 C C 


Sulphur dioxide 


- 10.1 


157 




lOnia 




131.9 


-77.7 


Car; :ide 


-78.6 


31.0 


-56.6 


Nitric oxide 


89,8 


36,50 


- 102,3 


Ethylene 


-in- . 


9,50 


-169 


ane 


-161.37 


2,85 


-183.15 


Carbon monoxide . 


-.190-0 


-13 




Oxygen 


- 182.95 


_m 


-218.4 


Nitrogen 


-19! 


-W 


-209,86 


. 


-245 


-228.71 


-2' 


Hydro 


-252 


-239.9? 


-259.14 


Helium 




-267 





16. Production of Low Temperature* by utilizing the Jonle-Thomcon 
Effect. — As the above method is not capable of liquefying hydrogen 
and helium, another process began to be utilized from 1898, This 
Is the Joule-Thorason effect discovered in 1852. A fall mathem 



VI. 



THEORY OF POROUS riAIG EXPERIMENT 



187 



analysis of the phenomenon is postponed to Chapter X. We 
describe the phfnomem n hare: 

We have already described Joule's eriment (p. which 

showed, that for permaneja the internal em tot depend 

upon volume, "I.e., t-jp | = 0. This is called Joule's law or Mayer's 

hypothesis and is the characteristic property of a perfect gas (for 
perfect monatomic gases U — ^NkT per rnoIA But Lids is not strictly 
true [or idle actual gases of nature ; they all show deviations from the 
st: te of perfectness and hence for them U is not independent of 
e. A slight change in the temperature of the gas should occur 
in joule's experiment, but since the capacity for heat of the contained 
air" is negligible as compared to the heat capacity of the surrounding 
, no change in temperature could be observed. In 1852 Lord 
Kelvin, in collaboration with Joule, devised a modification or Joule's 
experiment in which very small changes in temperature produced by 
expansion could be easily measured. This is called the "Porous plus 
experiment and provi an unfailing test of Mayer's hypoi 
Widi its help we can easily find how far a gas deviates from the state 
of being perfect. Before proceeding to describe this experiment we 
shall discuss the theory underlying it. 

17. Elementary Theory of the Porous Plug Experiment.-— In this 
experiment a highly compressed gas is being continuously forced at 
a constant pressure through a constricted nozzle or porous plug. The 
as its name implies", consists simply of a porous material, say, 
cotton-wool, silk, etc. having a number of fine holes or pores u 
thus equivalent to a number of narrow orifices in parallel. TT 
during its passage through tlie pores becomes throttled or wire-drawn, 
viz., molecules of the gas are drawn further apart from one another 
doing internal work against molecular attractions. This is always the 
case whenever a fluid has to escape through a partly obstructed 
passage. On either side of the plug constant pressures are maintained, 
the pressure on the side from which the gas flows being much greater 
than on the other side since the plug offers great resistance to the 
flow of the gas. This expansion is of a character essentially different 
from Joule's e ision. In Joule expansion, the gas expanded with- 
out doing any external work. Here it expands against a constant 
externa! pressure and hence it has to do external work also, together 
with any internal work, while some work is done on the gas as well, 

plug is surrounded by a non-conducting jacket so that the process 
is adiabatic in the sense that no ""heat enters or leaves the system. 
For such 'processes we now proceed to show that the total heat function 
h = u -{- pv remains constant. 

To prove this theorem, let us consider a. mass of the fluid 

ous plug C from left to right as indicated in Fi 

.. ,-.. 7" A , u A and p E} v Bf T B , u a be the pressure, volume, 

temperature and internal energy of one gram of the fluid before and 

after traversing the en":; re: respectively. Suppose that one gram of 



138 



PRODUCTION OF LOW TEMPERATURES 



I CHAP. 




L 



Vh 



C 



-•/.: 



M 



the gas is contained between the porous plug and some point. 

M on the left and also between 
the plug unci some point N on the 
right- For visualising the process 
we may assume an imaginary 
piston A at M separating this 
quantity of the gas, and that the 
flow of the gas is caused by the 
{or ward motion of the piston A. 
Actually, however, the rest of the 
gas exerts a pressure p± at M 
which is maintained by the source 
of supply. The gas after travers- 
ing the plug pushes forward the 
imaginary piston B whose motion 
is opposed by the pressure p of 
the gas to the right of B. The 
initial and final states are she 
respectively at (a) and (b) in the 
figure, the initial volume of the 




Fig. &,— The Porous Plug 
Experiment. 



.gas being equal to ML and the final volume equal to ON, 

The gas during its passage through the orifices in the plug has 

to overcome friction, viscosity, etc., and hence loses energy. The 

escaping gas issues in the form of eddies and its temperature falls 

cor ily just at the jet (and this effect is spurious) because some 

th'. i rgy is now converted to tile energy of mass motion. The 

x, subside after traversing a short distance and the 

.i.ture consequently rises. Let us consider only the steady 

and after transmission through the orifice, i.*?.., at points 

he plug where eddies are not present. We assume 

lowly so that the energy of mass motion 

is very sural! and negligible in comparison t nergy of thermal 

motions. Now our initial system is ML and the final system is ON. 

But the plug OL is initially and finally in the same state ; hence 

the change simply consists in a change from MO to LN* If in 

addition, the tube is surrounded by a non-conducting material, no 

heat is supplied to or withdrawn from the system. Some work is, 

however, done by the external forces on the slowly moving system. 

The force acting at A is equal to p A )< S where S is the cross-sectional 

area of the cylinder. The work done by this force upon the 

i: ' ;' } a)< 5 X MO = p x v A . Similarly, the work done by the gas in forcing 

the piston B is p^v*. Therefore, the net work done by the gas is 

put's — PaV*> and from the first law of thermodynamics, since 

A(> — 0, tile work done by the system is equal to the decrease in 

its internal energy, viz., 

»a -"b=Mb^Ma (i) 

OT *A +J*A»A = »S+J&B»B .. (2) 

Hence u -f- pv remains constant in the throttling process. 



"0 



THE POROL'S PLUG EXPERIMENT 



139 



For perfect gases Boyle's law flw) T ^ constant and Joules law 
u^=cT hold true, and therefore u-\-pv would depend upon tempera- 
cure only. We have just shown that u+pv for the porous plug experi- 
ment is constant whether the gas is perfect or not. Hence, tor 
perfect oases the temperature on both sides of the plug would remain 
the same. In actual experiments, however, a cooling- effect was 
observed £or most gases such as air, 2 , N 2 , CO, and a heating effect 
Thus none of the gases examined was peiletf. inc. 
be due either to deviations from Boyle s 
both* If we know the former, 
(p. 97) we am find the latter 
experiment. If at the particular 
pressure and temperature 1 1 A, the gas is more compressi 



than 

<*);. 
alone 
;as is 
Upon 
from 
work 



in case of Hj. 
lack of per redness may 
law or from Joule's law or 
as from Amagat's experiments 
by performing the porous^ plug- 
pressure and temperature in A. 

at lower pressures [c£, curves of N* C0 2 betore the bend ( 
& z?* <PuVn. Therefore, due to deviations from Boyle s k 
ut <ii i-e. the gas would show a cooling effect*. IT the 
l?ss compressible [cf., H a ) there would be a heating effect, 
these effects will be superposed, the effect due to deviation 

Toule's law. Since in actual gases, cohesive forces are present 

will be done in drawing the molecules further apart during expansion 

and the gas would become cooled. Thus the jouleThomson effect 

due to this cause would always be a cooling effect. Ihe observed 

effect is the resultant of these' two effects and may be a heating or 

cooling effect depending upon the sign and magnitude of the former 

effect/ 

18. The Porous Plug Experiment,— We shall now describe 

actual experiment. Joulef and Thomson 

were the first to carry out these experiments. 

They employed a cylindrical plug, which is 

indicated in' Fig, 9. The compressed _ gas 

Hows through a copper spiral immersed in a 

thermostat and after having acquired its 

temperature, passes through the porous plug 

W* The plug consists of silk, or cotton-wool 

or other porous material, kept in position 

between two pieces of wire-gauze and enclosed 

in a cylinder of some non-conducting wood 

bb. The plug and part of the tube is sur- 
rounded by asbestos contained in a tin cylin- 
der zz so that no heat reaches it from the bath. 

Joule and Thomson worked with air, 2 , N 2 . 

CO,, between 4° and 100°C, the initial and 

final pressures being 4.5 atmospheres and 1 
atmosphere respectively. 




Joale-Tnom- 
soh's porous pirns; 



of two 

energy 



'♦It is important to remember that the internal energy is made up 
parts ' ) kinetic energy which depends upon the temperature, (..•- 1 
due to molecular attractions, the former being much greater in a gas uik 
ordinary conditions so that Joule's law is approximately obeyed. 
{ foalej Scientific Papers, Vol. II, p. 216. 



140 



PRODUCTION OF 1X.AV TEMPtiRAi 



CHAP. 



Some of the subsequent worl^ers employed a plug of the "axial 
flow" type as used by Joule and Thomson, while some others 
err, i plug of the "radial How" type. In the latter the gas 

flowed from the outer side of a hollow cylindrical plug to the interior 
and hence heat insulation was better. Certain others employed only 
a throttle valve or a restricted orifice. 

From these experiments Joule and Thomson found that the fall 
in temperature was proportional to the difference in pressure on the 
two sides of the plug, i.e., bd—k(}> A —f! u ), where k is a constant, 
characteristic of the fluid. They found empirically that k — A/T* 
where T h the absolute temperature of the gas, while Ross-Innes 
found k = A -I- B/T, Hoxton, however, found that his results were 

represented by the form. 



£_.4+-j'~+^j 



(*} 



:]-?-0 



We can find a value of k from theoretical considerations i 
Chap. X) , If the gas be supposed to obey van der Walls' equation 
of state, it. can be shown diat the Joule-Thomson ef r 

Exercise .[.— U-; dues of a = 1-36 X MP atm - cm * an d & = 

32-0 c.r. for a gram-molecule gen at N.T.P. and C* = 703 cal. 

akulate the Joule-Thomson effect from equation (i) , 

9 = I_ -0 1 x 10* A 

Ap 7.03x4.18xl0 7 L 8< ^73 

; 4. 18x10' d 

01 X 10* _dcg. 

7*03x4.18x10* atrm/cms" 

1 r a van der Waals' gas the cooling 

duced in the Joule-Thomson process. 

I 11 * •' by the gas— 1 * ^dV = — — — , 

, • . Net work (external -[- internal) done by the gas 



pro- 



" 



=^f7"i-r 1 )+A( J fr i - A ) + 



::■■■ 



2a 

since from van der Waals' equation /?F — RT 4-bp — (a/V) approx 
lor the two sides of the plug. 

will produce a cooling by _ &q (since AC? - 0) such that 
(see Chap. X, sec. 10) - 



VL] 



PRINCIPLE OK KPGENK.RATIVK COOLING 



14? 



Wb'jt 



f ^fe 



or -C B AG - #A0+*AH-£j= (—Art a PP 

Combining this i b on C„ - C B t= R, we get equation (4) . 

19, Principle of Regenerative CooUns-— The Joule-Thomson « 

in s observed tor most gases is v. as, £<w a jr at a tem " 

oerature 20°C when the p iun on the two s.des are SU 
atmospheres and 1 atmosphere respectively, Joule and Thomson found 
that" ike tei;. re falls by Il-VC. Hence the method eduid not 

be employed for a long time tor producing liquefaction. Subsequent 
however, it was discovered that the cooling effect can be unengi! 
by employing what is called the reg ■ A portion ol 

tne eas which first suffers Joule-Thomson expansion and becomes 
mp] wed to cool other portions of the incoming gas betore 
die la- bes the nozzle. The incoming gas becomes still m * 

cooled after traversing the nozzle. In this way the cooling eltect 
can be made cumulative. In actual practice* this is secured by using 
either concentric tubes as in Linde's process or by means of Hampson 
spii r : i . Two or more concentric tubes are arranged in tlie 

form of spirals, die inner one carrying the high-pressure inflow 
gas while die outer one die low-pressure outflowing gas. In the 
regenerative method a further advantage 
is gained by the fact that the lower the 
ture the greater is the Joule- 
Thomson cooling. 

The regenerative principle is illustra- 
ted diagram matically in Fig. 10 where the 
high- pressure gas from the compi 
enters the spirals contained in the water- 
cooled jacket A. The gas next enters the 
regenerator coils at E and by expansion at 
the valve C becomes cooled by a small 
amount. This returns by ,thc outer tube 
abstracting heat 'from the high pressure 
gas, and reaches F almost at the 
temperature as the incoming gas at E. 
The As is again compressed and cooled by 
A and re-enters at E. As time passes, the 
gas approaching C becomes cooled more 
and more till the Joule-Thomson cooling 
at C is sufficient to liquefy it viz., its 
temperature reaches the value at which 

the gas would liquefy under the pressure prevailing at F. A portion 
of the esc::: i j gas then condenses inside the Dewar flask D. At 
this stage the temperature throughout the apparatus becomes steady 
i . ic represented by the curve shown in Fig-, 11, p. 142. The 
part L M" represents the continuous decrease of temperature of the 
as we approach the nozzle through the inner tube while MN repre- 
sents die Joule-Thomson cooling. NL represents the temperature 
of the low-pressure gas which is less than that of the adjacent high- 




f -:i; i ;l 




Fig. 10. — Illustration of the 
Regenerative Coding. 



142 



i'JiODUCTION OF LOW TEMPERATURES 



CHAP, 



pressure gas. Thus the cooler low-pressure gas abstracts heat from 
die incoming stream. 

20. Linde's Machine for 
liquefying Air,— The principle 
of regeneration (applied to heat- 
ing) was discovered by W. Sie- 
mens in 1*57, hut its application 
to cooling' processes came later. 
Linde* in Germany and Hamp- 
son in England almost simulta- 
neously (1895) built air lique- 
fying machines based on the 
above priii rip J e. At the present 
time, such machines have be- 
come quite common and many 
ies in tire world arc fitted up with such machines. Fig, 12 
3 a commercial form of Linde's machine; e, d, is a two, or better 




]}Js!anc« along (he Mgciicmlve tail 

1ml\ II, — Temperature distribution 




1 ... 12.~Linde's apparatus for liquefying air, 

three stage compressor, the machine e compressing the gas from 1 to 
20 atmospheres while d compresses it from 20 to 200 atm, A charge 
oJ atmospheric air is taken in at. e and compressed by e to 20 atm. 

* K:ir ' y» Linde, born in 1842, was Professor at Munich. He published an 

account of bis air- 1 . machine in 1895. 



VI.] 



AIR LIQtttiFIERS 



143 



then cooled by passage through water-cooled tubes and is ^ 
to the suction side of the second stage compressor. The tomprebsed 
L d!en pa^es tough the cylinder f which contains caustic soda. 
J his a^o?b tie carbSn dioxide (IE this is not done, carbon dioxide 
ifflbSS? Nidified and choke the valves in the ^^Hggg; 

■ -as then pusses to the tubes g which are cooled by a freeing 
m ixttne to . -20'C, From here it, passes on through the metallic tubw 
p^hiner coils of the liquefier proper. At a we have got the plug 

<in the first stage to 20 aim.), tb* temperature falls to about -<B <£ 
ind the air agali passes through the outer cods cooling the incoming 
t»- it is then led Sirowh the pipe P, to the second stage compression 
fnder where it is again compressed and allowed to pass through 
X reftigerator and thl inner coils to a. After the ^p etion o a 
few cycles the temperature of the incoming gas falls so low that the 
xerond throttle valve is opened The air .s now allowed to exp tndto 
1 atmosphere when it becomes liquid and ^ ct %^j^^ .: 
from which it can be removed by the siphon A. The unliquefied gas 
is aeain led back through the outermost cods to the compressor e as 
nSd by the arrowt Fresh charge of air is beifg continuou 
ta^en in at*, compressed and delivered along with the gas from the 
middle tube to d. The process is cyclic. 

21. Hampso^s Air Liquefier. This liquefier also lujlizes the 
joule-Thomson effect and the regenerative principle, but differs i from 
Linde's apparatus in details of construction. The special feature 
about it is the Hampton spiral. The high pressure inflowing ^as 
Passes through copper* tubes coiled in the form of concentric spirals 
arranged in layers; it then suffers the throttle expansion and becomes 
cooled This cooled air vises through the interstices between the 
a era or spiral and thereby cools the incoming li^h-pr^iire gas. 
After some time the high-pressure gas becomes sufficiently cooled so 
that on suffering the throttle expansion, it liquefies. The apparatus 
thus differs from that of Lmde only in the manner of cooling the m- 
comine 2as. The apparatus was later improved bv Olszewski. 1 he 
ITampson construction has been utilized later by Dewar, Onnes and 
Meissner for liquefying hydrogen (see Fig. 14) and helium. 

22. Claud* Air Liquefier*- Although the Linde and Hamp: 
lynchers just described are in extensive use in laboratories and com- 
mercial installations, the machines cannot be said to be satislactoi 

ainly because the efficiency of the machine, t.e. f beat extracted/ 
wiergy consumed bv the machine, is extremely low (about 15%). llie 
coolinir process employing Joule-Thomson expansion is really very 
tent. A more efficient machine could be devised if the com- 
,. K as was made to expand adiabatically doing external work 
and thereby suffered cooling, The technical difficulties in construct- 
,i apparatus for continuous liquefaction of gases by adiabatic 
expansion were overcome by Claude. The main difficulty consists 
in finding a suitable lubricant for the moving parts of the expan- 
sion cvlinder since the ordinary lubricants become solidified at these 



m 
en 



144 



PRODUCTION OF LOW TEMPERATURES 



[CHAP. 



low temperatures. 



C 




d) Low Pressure Air 



Fig. 13. -Claude-Heylaiidt s;. 



laude utilized petroleum ether as the lubricant. 
This ... i . iscous at tempera- 

Canvm* tures of - 140*C, or even - 160*C 

and thus acts as an effective lubri- 
cant up to this range. 

13 shows diag i i 
the Claude's air liquefying: 
chine, The gas (ram the com 
5 divided Into two parts 
to the expau- 
cylinder and suffers adiabatie 
onsequent coo] 
In this expansion it does 
work which is utilized in doing 
work on the compressor. The 
:d gas traverses upwards in ihe 
pipe D thereby coi i he second 
n of the Incoming com : 
in the second heat-exchanger. 
The high-pressure gas thus parti- 
ally liquefies. It then suffers Joule- 
Thomson expansion at the throttle' valve. The evaporated r 
taken to the compressor and afi 

Theoretically ( I ulc * be more efficient than the 

only slightly so. This is 

ilties of • ide method, The 

1 has the great advantage that no movable parts of the 

ad hence the csmtructia 
dt slight- Lqucfier. 

U ; taiuion turbine will po- vera! 

dprocariug engine, Kapitza in 1939 developed 
ier in which the compressed air is allowed to expand 
ire of abc ; ressure of about 1-5 atmos, 

and drive a turbine wheel, and thereby suffer cooling on account of 
ie. The machine is about three times as efficient as 
Lmde's li is it utilizes a pressure of about 5 atmospheres 

only, all danger due to high pressure is eliminated. 

23. Liquefaction of Hydrogen.— The method of cascades failed* 

to liquefy hydrogen, Wrohlewski thereupon studied the isotherms 

i gen at low temperatures, ai that caL the values 

and b, and thence the critical constants (p. 95) . The critical 

1. unci to be very low (— 240°C.) and Olszewski 

, ted out that this temperature could not be reached by the evapo- 

n of liquid nitrogen, which was the most intern 

shen known. The physicists then turned to the Joule-Thomson 

This !• however, first appeared to be inapplicable to 

Regnault had shown that when hydrogen 

♦Neon can be utilised but it is very rare. 






VI. 



HYDROGEN LIQUEFYING APPARATUS 



H5 



is subjected to this process, it gets heated instead of being cooled. 
That the difficulty is not fundamental and insuperable can be seen 
from the expression for the Joule-Thomson effect viz. } 

• . ■ ■ (4) 



Ap-CART } 



It is thus proportional to 2a/RT-b but in hydrogen and helium 
a is so small that at ordinary temperatures 2a/RT<,bj and the term on 
the right-hand side becomes negative. Hence, for a negative value 
of Aft *"■*» expansion of the gas, &Q is positive at ordinary tempera- 
tures, and the gas shows a heating effect. 

If T be sufficiently reduced, the right-hand term in (4) eventually 
becomes positive and the gas shows a cooling effect. There is just 
a temperature where 2a/R.t- b — Q, i\fi v where Joule-Thomson effect 
changes sign j this temperature is called the 'temperature of inversion' 
T-, which is thus equal to 2a/ bR. This relation gives, after substi- 
tuting the values of a and b for hydrogen, T,- to be - 73 1> C Olszewski 
experimentally observed the Joule-Thomson effect for hydrogen at 
various temperatures and found the temperature of inversion to 
be - 80.5 fl C. 

Similar is the case with all gases. There is a temperature of 
inversion for all of them which, however, depends upon the initial 
pressure- of the gas. Even the Joule-Thomson effect, depends very 
much upon the initial pressure. It is thus dear that the behaviour 
of hydrogen and helium is not anomalous ; they differ from other 
gases only in having a low temperature of inversion. 

Now since T t » the critical temperature, is equal to da/27hR 
(p, 95) and Ti = 2a/bR f it follows that T i= (27/4) T t . This rela- 
tion is found to hold true approximately. 

Hydrogen must, therefore, be cooled below ~80 D C, for liquefac- 
tion. But "for practical success it should be pre-cooled to the Boyle 

point T B (p, 88) which is defined as the point at which — ^-— — 0. 

Calculation with the help of van der Waals" equation shows that 
this temperature T B =a/bR r Hence r B = £7V We thus see that 
hydrogen should be pre-cooled to about 96°K {- 177'C). This tem- 
perature in easily attained if we immerse the hydrogen liquefying 
apparatus in a bath of liquid air. 

24. Hydrogen Liquefying Apparatus. — Dewar first succeeded in 
liquefying hydrogen in tliis manner in the year 1898, Travers later 
i 1 1 1 1 i : /red the apparatus. Hydrogen prepared from zinc and sulphuric 
acid is compressed to about 150 atrn, and then passed through coils 
immersed in water in order to deprive the gas of the heat 
Of compression. Next it passes through cylinders of caustic potash 
dehydrating agent and is deprived of its carbon dioxide and 
moisture. This is essential as these impurities would solidify much 
before the liquefaction of hydrogen sets in and choke the tubes. The 

10 



146 



l'RQDUCTION OF LOW TEMPERA! L RES 



[CHAP, 



gas then enters the liquefier and traverses the regenerative coils A 

(Fig. 1-3) which are cooled by the 
outgoing- cold hydrogen gas, and in 
the final steady state, becomes 
cooled to about — I70°G. Next 
the gas passes through a refrigerat- . 
ing coil B immersed in liquid air, and 
then through another refrigerat : n .; 
coil C immersed in liquid air boil 
ing at a pressure of 100 mm. This 
is adjusted by allowing liquid air 
from F to trickle into G and by eva- 
cuating G through a pump attached 
ai r. The tempera tine of the iiv 
drogen gas thus falls to about 
-200°C. Alter this it traverses the 
coil D and suffers Joule-Thomson 
expansion at the valve a which is 
operated by H. The gas thus be- 
comes cooled and this cold gas- 
passes up round the chambers G 
and F, thereby cooling the coils D 
and C, to the chamber R and from 
there to the compressor. Thus ■ 
alter a few cycles the temperature 
of the incoming gas at a falls to 
-25Q D C, and then on suffering- the 
Joule-Thomson expansion it liquefies 
and drops as liquid into the Dewar 

! I V. 




Fig. 14. — Hydrogen liqw 



Many later investigators devised apparatus which have a large 
output. Amongst them may be mentioned Nernst, Kamerlingh 
Onncs and Meissner. Dimes' apparatus h very similar to his helium 
liquefier. Meissner's apparatus is somewhat different in construction 
but similar in principle. 

Liquid hydrogen boils at -252.78 C C, under atmospheric pressure. 
By causing it to boil under reduced pressure it can be frozen to a 
white solid. 

25. Liquefaction of Helium. — Helium could not be liquefied 
for a long time. The attempts of Dewar and Olszewski to liquefy 
helium by the adiabatic expansion method were unsuccessful 
Kamerlingh Onnes,* however, proceeded in his efforts very systemati- 
cally. He studied the isotherms of helium down to liquid hydrogen 
temperatures (up to— 250°C) and obtained the critical constants for 
helium. He found the following values :—T { = 5.25'K, p ( = 2.26 
aim. and normal boiling point — "4.2f>°K. The joule-Thomson jnver- 

*He£ke Kamerlaiifih On ( 53— — 1926), bnrn in Holland, became Professor 
of "Physics at Leiden where he established his low temperature laboratory and 
investigated the properties, of substances at low temp 



vi.J 



SOLIDIFICATION OF HELIUM 



M7 



sion point came out to be about •55°K. and the Boyle point 17 :: K. 
is temperature could, therefore, be readied by pre-eooling the gas 
with liquid hydrogen. Kamerlingh Onnes: was thus convinced of the 
possibility of being able to liquet)' helium by the Linde process. He 
u actually liquefying it in 1908 in his laboratory ac Leiden. 
Subsequently helium liquefiers were constructed at Leiden, Berlin 
and Toronto. To-day there are scores of helium liquefiers in the 
world. Since helium is rather costly, the arrangement should be 
such that it can work in cycles, In the apparatus used at the 
t pyogenic laboratory at Leiden gaseous helium compressed to 36 atm. 
is passed through spirals immersed in liquid hydrogen boiling under 
reduced pressure and then through outgoing cold helium vapour. 
The gas then suffers Joule-Thomson expansion and becomes liquefied. 
The plant for liquefying helium is, therefore, complicated by arrange- 
ments tor liquefying air and hydrogen. 

Both hydrogen and helium were liquefied by Kapitza in 1934- by 
the Claude-Heylandt method. Helium was liquefied by Simon by the 
adiabatic expansion method and also by the desorption method. 
Collins in 1917 developed a commercial type of helium liquefier based 
on the Kapitza method. In the Collins expansion engine the piston 
and cylinder are constructed of nitrided nitralioy steel, the clearance 
being about 0.0005 inches on the diameter and the operation being 
completely dry. Thus the leakage of gas is extremely small and 
whatever does leak, also goes to the suction side of the compressor. 

26. Solidification of Helium. — Kamerlingh Onnes tried to solidify 
helium by boiling it under reduced pressure, but though he claimed 
to have reached 1.15 n K in 1910, helium still remained a fluid. In 
1921 he again tackled the problem and by employing a battery of 
large diffusion pumps he reduced the vapour pressure to 0.01.3 mm. 
and the temperature to 0.31°K,but helium still remained fluid. After 
the death of Onnes, his collaborator and successor, Dr. Keesoni suc- 
ceeded in 1926 in solidifying helium by subjecting it to an enormous 
pressure, Helium was compressed in a narrow brass tube under a 

fressure of 130 atmospheres, the tube itself being immersed in a 
[quid helium bath. Jt was found that the tube was blocked indicat- 
• that part of the gas had solidified. If the pressure was reduced 
I or 2 atmospheres, the tube became clear again. Later expen- 
ds showed that helium at 4.2*K solidified at 140 atmospheres 
while at 1.1 °K it solidified only under 23 atmospheres. Solid helium 
i be distinguished from the liquid : it is a transparent mass 
having almost the same refractive index as the liquid. 

27. Cooling produced by Adiabatic Demagnetisation. — Upto 1925 

tiethod available for producing temperature lower th 
is the boiling of liquid helium under reduced pressure. 
mi in this way reached 0.72°K in 1932, In 192G Debye and 
u theoretically that lower temperatures could be pro- 
by the adiabatic demagnetisation of paramagnetic substances 
(i.e. those substan ces for which the magnetic susceptibility y 
' i i . The principle of the method is as follows ; — 



M8 



PRODUCTION OF LOW TEMPERATURES 



[CHAP. 



The process of magnetising a substance involves doing work on 
it in aligning the elementary magnets in the direction of the external 
field. If a substance already maguc sei demagnetised adia- 

batically, it has to do work and the energy to do this work is drawn 
from within itself, in consequence of which it cools, This cooling 
can be made large if a .strong magnetic held is employed and the 
initial temperature is low because then the magnetisation produced 
in the substance is large. This follows from Curie';; law. which states 
that the paramagnetic susceptibility of a substance varies inversely as 
i u absolute temperature i.e. % = C/l\ The final temperature attain- 
ed is determined by measuring the magnetic susceptibility of the sub- 
stance and calculating from Curie's law. In this way de Haas and 
Wiersma succeeded in reaching the present low temperature record 
of about (UK)34° K in 198 > idiabatkally demagnetising mixed 
crystals of chromium-potaawum alum and aluminium-potassium alum 
at' 1.29°K from an initial field of 24000 gauss. 

28. Properties of substances at liquid helium temperatures. — 
Properties of substances undergo very interesting changes at ex- 
tremely low temperatures. K. Onnes found in 1911 that at abom 

metals appear to lose completely their electrical i 
and become superconducting. The resistance does not absolute 
vanish bin falls bo aboui a millionth of its value so that if a current. 
be induced in a toil ol Lhe metal placed inside such a low ten 
bath by bringing a magnet near it, the current does not in 
die out as in ordinary ele« ti aetic induction, but may continue to 

together. It. has also been found thai near the absi 

to vanish. Besides, liquid helium 
about 2 esses strange properties, the most im- 

portant being the property of superfluidity when the liquid has no 
viscosity. 

29. Low Temperature 
Technique* — Dewar ft-.i- 
The discovery of Dewar flask 
by Sir James Dewar* in the 
Royal institution of Loi 
provided a very convenient 
apparatus for low tempera- 
ture storage. Though very 
low temperatures had been 
produced, it was d:: 
to maintain the liquids 
at these temperatures as 
even by packing -the bottles 
with the I: 

material, the leakage oE heat, from outside could not be prevented. 

But the problem • ed by Dewar in a very ingenious way, 

* James Dewar (1842-1523), born at Kincardine, became Pn iess c oJ b 
Philosophy at Cambridge In 18/5 n,nd also Pi • n the Royal 

Institution in 1877, His i rfc was in the low temperature region, 




15,— Dewar Flask, 



VI.] 



USES OF LIQUID AIR 



149 



The Dewar flask (shown in Fig. 15 together with a siphon) con- 
sists of a double-walled glass vessel, the inside walls being silvered. 
The air is completely evacuated from the interspace between the walls 
which is then sealeti If some substance be now placed inside such 
a vessel and the top closed, it is perfectly heat insulated, except for 
the small amount of heat which may creep in by conduction alone 
the sides. The silver coatings protect the inside from radiation, and 
the absence of air prevents the passage of heat by conduction through 
the walls. If the substance be hung by thin wires inside the flask 
and i,l le latter evacuated by pump and sealed, the insulation is com- 
plete. Such an arrangement was used by Nernst in his low tem- 
perature calorimetry (p. 42) . Dewar flasks are now sold in the 
market under the trade name "Thermos flask". They have lately 
been made entirely of metal with a long neck of some badly con- 
ducting alloy as (ierman silver. 

Low Temperature Siphons. — For transferring liquid air from one 
vessel to another, special types of siphons are used. One such siphon 
is shown in Fig. 15, connected to the Dewar flask. It is formed of 
a double-walled tube silvered inside, the space between the walls 
being evacuated. On the application of gentle pri • -sure to the rubber 
compressors A or B, liquid air rises up trie siphon and can be trans- 
ferred to a second vessel. 

tvyostats. — For low temperature work constant temperature 
baths are necessary; they are called cryostats. The substance to be 
investigated is kept immersed 1 in these baths. From Table 3, (p. 136) 
it is easy to find out which Liquids are suitable in a particular range 
of temperatures. In this way suitable liquid baths can be easily 
constructed down to ~218°C/ When no suitable liquids are avail- 
able, vapours of liquids can be employed. 

30. Uses of Liquid Air and Other Liquefied Gases. — The import- 
ance of liquid air is being increasingly felt so much so that it has 
now become essential for several purposes. Bottles of liquid air 
can now be obtained in any important modern town at a compara- 
i ill cost. We shall give some of the important uses to which 
liquid air has been put r 

(') Prod of High Vacuum, — High vacuum can be obtained 

by using Liquefied gases with or without charcoal. For instance, if a 

1 1 1 fust fdled with a less volatile gas than air, say sulphurous 
i' id h water vapour, and is then surrounded by liquid air all the gas 
inside becomes solidified and thus high vacuum is produced. If the 

I contains air, liquid hydrogen may be employed to condense 
ii. I Ilia process is greatly assisted by charcoal which possesses the 
rem property of occluding gases at very low temperatures and 

the |i ; temperature the greater is the adsorption. Further also 

•he ion is selective; as a general rule it may be said that the 

more vola lie the gas, the less it is adsorbed. 

shall give a numerical example. During a certain experi- 
i' i in a vessel containing air at a pressure of 1.7 mm. at 15 a C, when 






l'UODUCTIOX OF LOW TEMPERA!!: •:.;• 



r.i-.\'. : - 



cooled by charcoal immersed in liquid air, gave a pressure of 0.000047 
mm. in an hour and using liquid hydrogen as the i oak sure 

was reduced to 0.0000058 mm. 

(ii) Analytical Uses of Air*— Liquid air is of great use in drying 
and purifying gases. Water vapour and the less volatile impurities 
are easily removed by surrounding- the gas in question (say H->) with 
liquid air, and for this purpose it is now used as a common labora- 
tory rea,:y 

(Hi) Preparation of Gases from Liquid Alr.~ Oxygt a k now pre- 
pared commercially from liquid air by fractional distillation. Since 
the boiling point of nitrogen is - 195,8 C C, and that of oxygen is 
- 182,9°*!, the fraction to evaporate first will be rich in nitrogen while 
-thai evaporating last will be rich in oxygen. A few fractional distilla- 
tions will suffice to separate these completely , Several r 

v been devised to effect this separation. In Linda's rectifier 
(1902) liquid air trickles down a rectifying column where it meet, i 
upgoing stream of gas. The temperature at the top of the column 
is slightly below -194°C, {B, P. of liquid air) while at the bottom 
it is - 183 °C. (B. P. of oxygen). 'Ihe rising gas at the bottom 
comes in contact with the down-coming liquid and thereby some 
oxygen of tire risinu condensed, while some of the nitrogen in 

the downcoming liquid evaporates, and the liquid also becomes 

oner. The process continues till the liquid reaches the bottom 

whi mains nearly pure oxygen, while nitrogen passes off as 

mi at the top. gen is almost pure but die nitrogen con- 

of oxyg <re efficient rectifiers have since been 

devised by other workers. For details see Separation of Gases by 

. Run em an n, Chaps, VI, VII and VIII. 

gain atmospheric air may be utilised for the production of the 
rare gases, particularly helium, neon and argon. Roughly five 
volumes of helium are found in million volumes of air but this is 
sufficient for our purpose. Liquid air may be separated into two 
fractions, the less volatile part consisting of O a , N 2 , A, C0 2 , Kr, Xe 
and the more volatile part consisting of He, H 2f Ne.^ Thus in the 
rectifier described above the gas going up will contain N tt , LL, He 
and "N T e. The nitrogen is removed by passing the gas through a 
dephlegmator and hydrogen is removed by sparking with oxygen. 
Neon and helium can be separated by cooling the mixture with liquid 
hydrogen. Thus, oxygen, nitrogen and helium may be obtained from 
. For details see Ruhcrnann, Separation of Gases, Chap. IX. 

(iv) Calorimelric Application*. — Dcwar constructed calorimeti 
of liquid air, oxygen and hydrogen. lie employed pure lead as the 
heater, and the volume of the gas evaporated by the application of 
this heat was measured. These calorimeters have the advantage 
that a large quantity of gas is formed which makes it possible to 
detect as. little as 1/S0O calories with liquid hydrogen. In this way 



V1 1 PRINCIPLES OF AIR CONDITIONING 

the specific heat of lead and other substances may be investigated 
at low temperatures. 

(v) Use of Liquid Gases in Scientific Research^-Jhe extreme y 

low temperatures which are now available to us by the use o£ liquid 

and ifquid hydrogen have opened for the investigator a new and 

vast field for research. Tins has made a liquid air plant essentia 

,,,;,, modern laboratory. Most of the important OT^""* 

matter have been investigated at low temperatures and have y, elded 

X of far-reaching importance. This has been extended even » 

biological research, where it has been shown that bacteria as well as 

their activity unimpaired even after exposure to liquid 

air temperatures though a moderately high temperature is fatal. 

(vi) Industrial Uses of Liquid G««s.— Liquid air is used com- 
mercially for the preparation of liquid oxygen as explained above. 
For submarines anS aeroplanes it may be found useful to store liquid 
air or liquid oxygen for respiration but the low thermodynamic effi- 
ciency inherent in the Uncle machine prevents the use of liquid air 



oxygen "is, employed on a small scale for preparing expk 

with powdered charcoal and detonated, it explodes with gieat 

violence. 

31. Principles of Air-conditioning.- The Comfort Ctort^-The 

casonal variations of temperature, humidity, etc, have marked effect 
m PTowth, longevity and working efficiency of man. The seasonal 
of the Near lead lis to change periodically our clothing, food 



on gro 

changes of the year lead . 

and manner of living. But we can hardly cope adequately with the 

variations unless we" can really control the weather changes within 

our comfort limits as regards temperature, humidity and other factors. 

The science of refrigeration, heating and ventilating devices have 

rendered it possible to control the weather at least within the tour 

wails of our room. This particular branch of study is known as the 

science of air-conditioning. 

Complete air-conditioning means the control of the following 

rs : . 

.Average comfort condition. 

75-7 7 'T. 

60-65%. 

25 75 ft./min. 

at least 25% of total circulation. 



I) 



Temperature 

Relative Humidity 

Air movement 

Introducing Fresh Air 

Purification of Air 

i ii ido tzing 

[<<iv Li i >r Activating the Air. 



Although apparently temperature seems to be the only guiding 

, i,„ ; Lng, the relative humidity (r.h.) plays almost 

ly imp 'le in the feeling of warmth. The same tempera- 



m 



PRODUCrriON OF low temperature? 



[chap. 



ture condition, say 75°F, may make us feel either a bit too warm 
or too chilly according as the r.h, is too high or too low. This is 
because the humidity condition controls the evaporation from our 
body and hence the abstraction of latent heat which gives rise to 
the different warmth feeling. It is interesting- to note thai, some 
of the air-conditioning plants in the tropics (Calcutta) do not employ 
any heating device during winter but only humidify the atmosphere 
by atomised spray of water. This is because the average winter 
temperature of the place (about 70 — 75°F indoors) is not 'really too 
low, but what makes us feel chilly is the low (40%) r.h. So we feci 
quite comfortable only by raising the r.h. up to 60— 70%. In cold 
countries, however, rooms are conditioned by electrical or steam-pipe 
heating devices* 

it is interesting- to find that the comfort feeling is fairly critical, 
that is to say, that individual variation does not go much oft' the 
average. Elaborate experiment? have been performed by the Harvard 
School of Public Health in collaboration with the American Society 
of Heating and Ventilating Engineers on the average comfort feeling 
under the different combinations of temperature, humidity and air 




Dlty BULB TEMPERATURE 
Fijr, 16.— The comfort chart. 

velocities, etc. As a result of these experiments the comfort chart 
(Fig. 16) is drawn in which the co-ordinates are tbe dry and the wet 
bulb temperatures and lines of constant r.h. and comfort scales for 




vi.j 



PRINCIPLES OF AIR-CONDI HOKING 



153 



cliU 







summer and winter are also drawn. The chart shows that 98% ol 
the people during- summer would feel very comfortable at 71°F. 
effective temperature. We shall see presently what effective tem- 
perature really means. It relates to the human feeling of warmth 
under various combinations of temperature, r.h. and air velocity. 
For example, this 7l°F effective temperature., may be obtained by 
various combinations, such as 40% r.h., 78°F dry bulb, 62°F wet 
bulb ; or 60% r.h., 75°F dry bulb, 60 D F wet bulb ; or 70% r.h., 74°F 
dry bulb, 67 b F wet. bulb (all with air velocities 15—25 ft./min.) . So 
we see that the term 'effective temperature' represents a new scale 
which enables us to standardise the comfort feeling due to the various 
combinations of dry bulb and wet bulb temperatures (which automa- 
ticalfv define r.h.) and air movements. The three combinations 
(starting with 40%., 60% and 70% r.h.) as exemplified above would 
give rise to a feeling as if the person were placed at a temperature 
of 71°F in a saturated atmosphere and with still air. This is how 
we can define effective temperature. 

The effective temperature scale thus represents the conditions 
of equal warmth feeling with various combinations of temperature, 
humidity and air move- ^ 

men is. These are find- 
ings based on experi- 
ments with men and 
women with normal 
clothing- and activity 
and subjected to condi- 
tioned atmospheres of 
the various combina- 
tions of temperature, 
r.h., and air movements. 

Fig. 17 shows the 
el i :: c rive tempera ture 
chart. Let us see how 
to read the chart to find 

the effective tempera- 
ture. Suppose that we 
have in trie room an 
atmosphere with dry 
bulb temperature 76°F, 
and wet bulb 62 D F and 
wind velocity "30 ft./ 
min r (0-30 ft./min, may be taken as good as still air.) Now put the 

rht edge of a scale across the two temperatures (the dotted line) , 
and it intersects the effective temperature lines at about 70°F. Tf the 
same temperature of 7G°F dry bulb and 62°F wet bulb are found to 
exist with a wind velocity of 200 ft./min. our feeling would be corres- 
ponding to (574-°F effective temperature and so on." 

We have seen that temperature and humidity are the most 
important factors for comfort feeling, while movement of air gives 




17.— Effective temperature chart. 






PRODUCTION OF LOW TEMPERATURES 



CHAP. 



a feeling of ease if ic has velocities within 25-75 Et/min. If the 

air is dead still, it becomes uncomfortable and stuffy. On the other 
hand, if it has too high a velocity it. becomes blasty and we would 
not like it. The higher the wind velocity, the colder is our feeling, 
as it facilitates evaporation from our body. 

We next consider fresh air. A room-cooler (or a room-heater 
in the cold season) must make provision for introducing sufficient 
amount of fresh air into the room. Of the total circulation a mini- 
mum of 25% fresh air is recommended, the remaining part is the 
room air itself, recirculating through the machine. It h true that 
it more air is drawn from outside and cooled in the air-conditioning 
machine for distribution in the room it would be better, but it becomes 
too expensive since much more work has got to be done in ord 
cool the large bulk of outside hot air. 

In a complete air-conditioning outfit, devices are included to 
purify and deodorize the air by suitable means. 

In spite of the complete air-conditioning arrangements it is found 
that we can never feel like the natural atmosphere under the same 
conditions. Recently it has been found that the amount of electri- 
cally charge ions present in fresh atmosphere is higher than that 
present in the air of an occupied room. Experiments have been 
made to ionise air by X-rays and introduce the ions into the room in 
portions to activate the atmosphere of the room. This has 
given positive effect. 

32. The Air-Conditioning Machine, — We have so far seen what 
iing actually in: Ve shall now consider how it is 

ved. 




Fig. 18.— The Evaporator. 



The air-conditioning machine or the room-cooler is fundamentally 
a refrigerating machine which has been described in section 7, p. 130 
with the main difference in the design of its evaporators. This is 



THE AlR-CONDlTlONlNG MACHINE 



155. 



VI.] 

just tl»e question of how we want to utilize the cold produced by the 
refrigerating machine. In air-conditioning unit the evaporator con- 
sists of a scries of zigzag copper-tubings thoroughly firmed with thin- 
copper sheets in order to get a large area of cold surface (Fig. 18) - 
As the liquefied refrigerant (SO*, Freon, etc.) evaporates in the tubing 




fier, etc, clearly 



HB& 



i**Wl«'«B»r.-*i.v:.. 




Fig. 19. — Frigidaire conditioner, 

It is important, to note that die summer air is laden with much 
moisture and it is desirable that humidity should be lowered. As 
the air is fanned through the cold fins of the evaporator the moisture 
condenses on them into droplets which are ultimately drained off. 
Thus cooling and dehumidification are simultaneously brought about 
in die same process. 



156 



PRODUCTION OF LOW TEMPERATURES 



[CH IP. 



The size and capacity of an air-conditioning machine is not 
determined only by the size of a room. It depends' upon the follow- 
ing considerations of heat loads ; 

(i) Sun's rays falling on walls or roof. 

(ii) Conduction through walls and roois due to the difference of 

outside and inside temperatures, 
(tti) Human occupancy. — For small private installations this heat 
load is not more than 5% of the total load but in cinema or 
theatre halls, it is 55 to 65% and in restaurant 40 to 60%. 
(Average heat dissipation is taken to be 400 13. Lu. per hour 
by each person) . 
(iv) Infiltration, Le. } outside unconditioned air entering through 
conditioning machine itself (its motor) , cooking stove, etc. 
(v) Heat-producing items in the room, e.g. tie lamps, air- 

conditioning machine itself (its motor), cooking stove, etc. 
In order to minimise the heat load which mostly enters from 
outside, the walls and ceiling must be covered with insulating boards 
such as celotcx, masonite, etc. and matting for the floor should 
he used. 



America and most of the European countries, But for tropical 

like India, the chart would differ considerably. People in 

the tropi omed to more warmth and humid atmosphere, 

an{1 tni ican air-conditioning- machines have got to 

readjusted according to our comfort conditions. 

Books Recommend* 

1. Andradi 

Z I,. C. Jackson, Low Temperature Physics (1950), 
Mediuen & Co. 

3. M. and B. Ruhemann, Low Temperature Physics (l°-§7) , 
Gam bridge University Press. 

4. C. F. Squire, Low Temperature Physic* (1953) , McGraw-Hill 
Book Co, 

5. M. Ruhemann, The Separation of Gases (1940) , Clarendon 
Press, Oxford 

fi. Glaze brook, A Dictionary of Applied Physics, Vol 1, articles 
on 'Refrigeration' and 'Liquefaction'. 

7, Mover and Fittz, Refrigeration, 

8, Hull, Household Refrigeration, Published bv Nickerson & 
Collins Co., Chicago. 

9, K. Mendelssohn, Cryophysics, (I960), Intersciencc Publishers, 
Inc., New York. 



CHAP PER VII 
THERMAL EXPANSION 

1, The size of all material bodies changes on being heated. In 
the majority of cases, the size increases with rise in temperature, 
the important exceptions being- water and some aqueous solutions in 
the range to 4 fi C, and the iodide Of silver (resolidified) below 
142°C. We shall first consider the expansion of solids. 

EXPANSION OF SOLIDS 

2, The cubical expansion of solids is somewhat difficult to 
measure directly (a method is given in section 17) , and is generally 
calculated from the linear expansion. Hence, experiments on die 
expansion of solids generally consist in measuring the linear expansion 
of bars or rods of the solid. For isotropic bodies whose properties 
are the same in all directions, the expansion is also the same in all 
directions. To this class belong amorphous solids (e.g., glass) and 
regular systems of crystals (e.g., rock salt). Metals may also be in- 
cluded because though they are composed of a very large number 
of small crystals, these crystals are oriented at random and the 
average properties are independent of direction. In anisotropic 
bodies such as many crystals, the expansion is different in different 
directions and may be even of different sign. We shall first consider 
isotropic bodies. 

ISOTROPIC SOLIDS 



3. Linear Expansion. — If a bar of length I, } at 0°C. occupies a 
length I: when raised to t a C. t l t can always be expressed by a rela- 
tion of the form 

where X is called the mean coefficient of linear expansion between 
and t°Q., and is a very small quantity. This differs very little 
from the true coefficient of linear expansion « at the temperature 



.' which is equal to 



1 dl 



I dt' 



The true coefficient a may also be defined 



] <!l 



by the relation a=-, — j~ which on integration will yield (1) if 
to at 

fadi — \l. The mean coefficient may be put equal to the coefficient 
of expansion at f/2°C if the range of temperature is small. Often 
the initial length is measured not at 0°C but at r_ L Q C. Then if t. z 
denotes the other temperature at which the lengh is l& we have 



158 



THERMAL EXPANSION 



[QIAP. 



Hence 






by the binomial expansion 

A = 5S=S approx 

The mean coefficient A itself is found to vary with temperature. 
This implies that the relation connecting- length and temperature is 
not a linear one and equation (1) must be modified into 

-where the successive coefficients go on decreasing rapidly. An equa- 
tion of this type is entirely empirical. The molecular theory of matter 
lias not yet been developed sufficiently to yield an exact 'theoretical 
formula. Generally it is sufficient to include terms up to the square 
of t ; the relation then becomes parabolic. In most cases both the 
coefficients Aj and Aa are positive, the body becoming more expansible 
as the temperature rises. 

4. Earlier Measurements of Linear Expansion. — The Linear expan- 
sion of solids is very small ; a bar of iron one metre long when heated 
from to 100°C, increases in length by about 1.2 mm." To measure 
such small changes in length accurately, special devices are necessary. 
The increase in length may be obtained from the readings of a sphero- 
m eter, or directly observed by means of a microscope. Again , the 
expansion may be multiplied in a known ratio by utilising the principle 
of the lever. The most satisfactory method, however, consists in 
utilising the interference fringes, which is considered in detail later. 
In this section, we shall consider the earlier experiments. 

The spheroiriLtcr or a micrometer screw was generally employed 

to measure the expansion and is suitable for ordinary work. The 

f. about a metre long, has its one end pressed against 

ted screw while the other end is free to expand. There is a micro- 

■ ■ spheroineter which can be brought into contact with 

his em noting the micrometer readings when the screw is in 

contact at 0°C., and at any other temperature (°C„ the expansion of 

the rod is found, whence the mean coefficient of linear expansion can 

be calculated from equation (1), 

Roy and Ramsden employed microscopes to measure the cxpan- 

□ and were able to obtain results of considerable accuracy. The 

experimental bar was placed horizontally in a trough between two 

standard bars, and parallel to them. One standard bar carries a 

cross-rnark at each end while the other carries at either end an eyi 

piece provided with cross-wires. The experimental bar carries an 

object glass at both ends so that the eye-piece on tire standard rod 

and the object glass on the experimental bar together formed a 

microscope focussed on the cross-mark on the second standard b: , 

The standard bars were always kept in ice. One end of the experi- 

ntal bar was fixed while the other end was free to move when the 

bar was heated. The object glass was brought back to its initial 

position by a fine micrometer screw, whose initial and final readings 

the expansion. 



ru.J 



COMPARATOR METHOD 



159 




1 . — Apparatus of 
Lavoisier. 



Laplace and Lavoisier employed the lever principle to magnify 
the expansion (mechanical lever method) , The change in length was 
converted into a change 
in angle by means of a 
lever arrangement and 
the angular change was 
measured by a scale and 
a mirror or telescope. The 
principle of their appara- 
tus is indicated in Fig. 1. 
One end A of the experi- 
mental bar AB is fixed 
while the other end B 
pushes against a vertical 
lever OB attached at 
right angles to the axis 
Of a telescope LL, which 

is itself pivoted at O and is focussed on a distant vertical scale CC'. 
The bar AB is first placed in melting ice and the scale division C 
■ en duo ugh the telescope is noted. Next the bar is enclosed in a 
hot-water bath. The rod AB expands to B' thereby tilting the 
telescope LOL to the position L'OL' and the scale division C is now 
seen through the telescope. The expansion BB' is equal to OB tan 
L BOB' = OB tan ./ COC = OB >( CC'/QC.. 

Paschen employed a combination of the micrometer screw and 
the lever. The expansion was multiplied in the ratio 1:5 by the 
lever and this magnified change in length was measured by the 
micrometer screw. An optical lever arrangement is also sometimes 
used when the expansion causes a plane mirror to tilt and thereby 
deflect a ray reflected from the mirror. 

5. Standard Methods. — At the present time the standard methods 
employed for measuring expansion are ; 

(1) Comparator Method, (2) Henning's Tube Method, (3) Me- 
thod of Interference Fringes, Methods (1) and (3) are direct while 
{2) is indirect. 

6. Comparator Method. — This is a standard precision method for 
■determining the expansion of materials in the form of a bar or tube. 
The bar, about a metre long, is mounted horizontally in a double- 
walled trough so that it can expand freely at both ends (Fig. 2) and 
has two fine marks L, L made near the ends. A standard metre is 

i mounted horizontally in another double-walled trough, both 
these troughs being arranged parallel to each other and mounted on 
rail: it either "the experimental bar or the standard metre can he 

>ughl into the field of view of two vertical microscopes M, M. The 
mi< . are provided with an eyepiece micrometer or can be 

moved parallel to the direction of expansion by means of a micro- 
meter screw, and are fixed vertically in rigid horizontal supports pro- 
jecting out from two massive pillars, the distance between the micro- 
scopes being about one metre. 



160 



THERMAL EXPANSION 



[CHAP, 



First the two troughs are filled with water surrounded by melting 
ice in the space between the double walls. The experimental bar 
is then wheeled into position and the fine marks on it arc viewed 














Fig. 2, — The Comparator method- 
through the two microscopes and their positions are noted in the 
micrometer. The standard metre is then brought below the micro- 
scopes and the two extreme marks of graduation are viewed through 
the microscopes. Front the change in the micrometer reading die 
length o£ the bar at 0°C is obtained. The experimental liar is then 
heated b\ replacing the melting ice in the double- 
walled space of the trough by water which is heated 
under thermostatic control. The fine marks are 
i through the micrometer eyepiece. 
Hie increase in length is determined from the 
change in micrometer reading. For measure- 
mem •. temperatures the experimental rod 

is placed in a tube which is immersed in a suitable 
liquid bath (e.g. liquid air) . 

7, Hearing's Tube Method of Measuring 
Relative Expansion.— In this method the ex- 
perimental and the comparison bodies are 
together brought to the same temperature and the 
differential change of their lengths is measured. 
The comparison "body is so chosen that its expan- 
sion in the temperature region is accurately known 
and, if possible, is also very small. Fused silica 
serves this purpose well. Inside a long vertical 
tube made of some well-defined glass (fused silica, 
Jena glass) , there is a ground point, molten and 
drawn out of the same glass at. its lower end. 
Upon this point rests the experimental ind R 
(Fig, 3) , about 50 cm. long and having both of its 
end faces ground plane. Upon the upper surface 
of R rests a pointed end of another glass rod S 
made of the same glass as the outer wider tube. 
To the upper end of this rod as well as of the 



S — _ 



worn. 



ft' 



C- ; i 



fM t 



*«*♦ 



Fig, 3, — Hetmiog's 
apparatus. 



outer tube are attached end-pieces carrying scales. The whole tube 



vn. 



FIZEAU S INTERFERENCE METHOD 



161 



up to half of the height of die rod S is immersed in a hot or cold 
bath and the relative shift, of the end-pieces is measured with a 
microscope provided with a micrometer eyepiece. The shift gives the 
relative expansion of the experimental rod against a glass tube of 
equal length. This is so on the assumption that the temperature of 
the rod and of the outer tube is the same at the same height. For 
high and low temperatures suitable baths may be employed. 

g. Fizeau's Interference Method. — Fizeau devised an optical 

method '.'< .; upon the observation of interference fringes. This 

method is capable of very great accuracy and is specially suitable 
when small specimens of the experimental substance are available, 
as in the case of crystals* 

In his original experiment, Fizeau used the substance B (Fig, 4) 
in the form of a slab about 1 cm, thick with two of its opposite plane 



X' 



t 



-..: 



M 





Fig. 4. — Fizeau' s interference method, 

rail el and polished. It was placed with one of these faces 
tl on a metal plate A supported by three levelling screws S, S. 

Vh projected upward through the metal plate a little beyond 

the upper surface of the slab B. A convex lens L having the Tower 
;• large radius of curvature was placed on these screws so 
tin i in film of air lay between this surface of the lens and the 
upper polished surface of the slab. With the help of a mirror M and 
a right ui| I d prism P placed above the lens, horizontal rays of light 
Erora ;i sodi im fl; ae F were sent down vertically to illuminate the air 
film and the i ays reflected repeatedly at the surface of the slab and the 
lower surface of the lens proceeded vertically upwards and were again 
reflected by the prism P and received by a horizontal telescope T so 
that Newton's rings could be seen through the telescope. 

11 



162 



THFRMAL EXPANSION 



CHAP. 



We know that in the case of Newton's rings the condition lor a 
bright ring is 

S = 2 t ie cos r « {In -f- 1) A/2, 
where 8 is the path difference, e the corresponding thickness of the air 
film and p its refractive index, r the angle of retraction of the ray into 
the film, n the order of rings and A the wavelength of the light used. 
In the present case ^ = 1, r = 0. We have therefore 

8 = 2*= (2n -f 1) A/2. .... (4) 
The difference in the thickness of the film at two successive bright 
rings is A/2, Hence when the thickness of the film changes due to 
expansion of the slab and of the three screws supporting the lens the 
rings appear to pass across a mark in the lens. Since one-tenth of the 
distance between successive bright rings could be measured, the change 
in length of the order of A/2 i.e. about 0.00002944 mm. could be 
determined. 

When the above arrangement, producing- the air film was enclosed 
in a chamber which was heated, the thickness of die film changed due 
to the differential expansion of the screws and the substance E, and 
the shift oF bright rings across the mark was observed. If x bright 
rings are thus shifted, the difference in the expansion of the projecting 
portion of the supporting screw and of the slab B along its cnickna 
equal to xX/2. In order to find the expansion of the screws* the slab 
was removed and interference rings were produced by reflections at the 
surface of the lens and the polished surface of the metal plate 
through which the screws projected. 

►be and Fulfrich improved Fizeau's apparatus by replacing the 
screws by quartz rings as shown in Fig. .5, p. 163. G and D are two 
quartz plates and R is a hollow cylindrical tripod, also of quartz, cut 
with its generating axis parallel to the optic axis, and placed b< 
G and D. Tin:- specimen is placed inside R and the fringes are formed 
by ' n-shaped air film enclosed between the lower surface of D 
and the upper surface of the specimen, the angle of the wedge being 
very small. The light from a Geissler tube (Fig. G) containing mercury 
and hydrogen is used. It enters the telescope at right angles, is 
deviated through a right angle by means of the prisms P, P and 
then falls upon the system as a parallel beam. The fringe systems 
for different wavelengths are formed at different heights, in the focal 
plane of the objective. By turning a screw any of these systems 
ran be brought in the field of view of the micrometer eyepiece. The 
lower surface of the upper quartz plate B is provided with a 
mark of reference and the number o[ fringes crossing this reference 
mark due to rise of temperature can be measured with die help of 
the micrometer eyepiece. If X lt A a} A a ,. ...... denote the various wave- 
lengths of light employed and Xi .-£■ § v .\,-H s , * a +6, the number 

of interference bands displaced across a fixed line (x representing a 
whole number and £ a fraction) , the increase A in the thickness of 
the ah' film is given by 



VII.] 



THE FRINGE WIDTH DU.ATOMETRR 



168 




n : ..- rings 

of Abbe, 



Fig. 6. — Apparatus for measuring expansion 
of crystals by Fizeau's method 



A= ^(*i+!i) - £ (*+*■)- | (*+*>}■ 



^ 



9. The Fringe Width Dilatoraoter. — In the last method the change 

in length was found from observations on displacement of the fringes. 

Priest devised a dilatomctcr in which changes in length can be 

obtained from the change in width of the interference fringes. The 

aratus is indicated in Fig. 1, p. 164. 

The air film is enclosed between the lower surface of the ro 
and the upper surface of the base plate, both of which are opti- 
cally plane and enclose a wedge-shaped space (0.1 to OS mm. thick). 
ample under test ends at the top in a fine point X xipon which 
rests the cover plate. On looking down in the direction OO, a system 
of interference fringes will be seen (as shown in the plan) appearing 
to lie in the plane bb so that the fringes and the referent 
and xx on die mirror can be simultaneously focussed. When the 
sample expands on heating, it tilts the cover plate and thereby 
thickness of the air film and consequently the width 
of the fringes'. The number of fringes between the lines ss and xx 
are observed both initially and finally, and from this the expansion 
can be calculated. 



1G-1 



THERMAL EXPANSION 



CHAF. 



rlr.:; 




The calculations can be readily 
made. We saw from equation (4) in 

last -section lliat if the film thick- 
ness increases by A/2, there is a shift 
o£ one fringe across the mark, the 
fringes actually contracting. Thus 
if in the present arrangement the 
number of fringes between the marks 
ss and xx changes by x-\-i), and d 
denotes the distance between ss and 
and A the wavelength of the light 
employed, then the change <t> in the 
angle between the planes bb and cc 
measured in radians is given by 

Again, if D is the perpendicular 
distance from X to knife-edge SS, A 
the relative expansion of the sample 
with respect to a piece of equal height 
made from the material composing 
the base plate, then ^ is also given by 



\— Fringe width ditatCHneter. 

6) and (7) we get 



*-*■ 






£)• 



CD 



(8) 



Knowing A» the coefficient of expansion can be calculated, 

10. Discussion of Results*— Table 1 gives die mean coefficient 

of. expansion of several substances between and 100 D C. The mean 

coel :!■ multiplied by 10 fi is given in the table, 

Table 1. Coefficient of Linear Expansion of Substances. 



Substance 


Ax 10 6 


bstance 


A >< *0 e 




pur °C 




per r C 


Aluminium 


: v.;.:-"; 


Platinum 


8,8G 


Copper 


16.66 


Palladium 


11.04 


Cadmium 


SIM 


Silver 


19.68 


Chromium 


8.4 


'1 ungsten 


4.5 


Lead 


28,0 


Brass 


18.9 


Magnesium 


26,0 


Invar 


0.9 


Manganese 


22. S 


Jena glass 


8.08 


Molybdenum 


5.20 


Pvrex glass 


3.3 


ickel 


13.0 


Quartz glass 


0.510 



v.q 



EXPANSION OF ANISOTROPIC BODIES 



165 



But as already mentioned in section 3, these values change 
appreciably if the final tempera tore is different from 100 ^C The 
mean coefficient A is a function of the temperature. As the final 
temperature is lowered the coefficient decreases. Grimeisen lias 

found the value of the quantity j ~ for very low temperatures and 

has deduced an important law connecting die coefficient of expansion 
and the specific heat. Gruneisen's law states that for a metal the 
ratio of the coefficient of linear expansion to its specific heat at con- 
pressure is constant at all temperatures. 
U. Surface and Volume Expansion,— The change in area and 
volume can be easily calculated from a knowledge of the coefficient 
of linear expansion.' A rectangle of sides / and b will, on being 
heated, have sides of lengths 1(1 +AQ and & (I + At), and its area 
will become lb (1 -j- Ai) 2 . If the initial and final areas be A and A 

we have , A , 

A — A (1 4- 2A1) approx., . . . (9) 

since A is small. Thus the coefficient of surface expansion is 2 A. 
Similarly, the coefficient of volume expansion can be shown to 

12. Expansion of Silica Glass, Invar.— Silica glass (quartz which 
has been -lidified into die non-crystalline form) is now 

: i v esipL or the construction ;of thermometers. The 

ansion of silica h very small (A = 0.5 X 10 -9 per °C.) , and is very 

con- ntl) determined by Film's method* Vessels made of this 
material can be heated without any fear of breaking. The curve 
connecting the coefficient of expansion and temperature is a straight 
line ihe room temperature and 1000°C. but at both limits 

it bends. The coefficient is negative below — 80 C C, 

Invar is another special substance, being an a!loy of nickel (fCyf ) 
and steel. Its coefficient of expansion at ordinary temperatures is ex- 
tremely small and hence it is generally employed for making secon- 
dary standards of length, and in the manufacture of precision clocks 
and watches. 

Anisotropic bodies 

13. Tt was first observed by Mitscherlich that the angles be- 
tween the faces of cleavage^ of a crystal of Iceland spar change when 
the crystal is heated. He gave the correct explanation of the pheno- 
menon, viz., that the expansion of the crystal is different in different 
directions and this is the cause of the change in angle. Such sub- 
ices are called anisotropic or non-isotropic. 

For every crystal, however, there can be found three mutually 
perpendicular directions such that if a cube is cut out of the crystal 
with its sides parallel to these directions and heated, the angles will 
remain right ancles though the sides will become unequal. These 
directions are called the principal axes of dilatation and the coefficients 
of expansion in those directions are called the principal coefficients 



166 



THERMAL EX P A MSION 



[CHAP. 



of expansion- Denote these by X x , Aj, A^- Then a cube o£ sides U will- 
on being heated to *°C, become a parallelepiped whose edges will 
be given by 



, =y*[l + (A* -M, +**)'] 



(11) 



The volume coefficient of expansion is thus A, +*«+**■ The Linear ex- 
pansion in any other direction can be readily calculated in terjns of 
the principal coefficients and the direction cosines.* 

14. Experimental Methods and Results,— Crystals are best investi- 
gated by the interference method. The crystal is cut in the manner 
desired, into a plate with parallel faces from 1 to 10 mm. thick, 
and is placed between the glass plate and the metal disc. The de- 
link of these experiments have already been given. 

When the expansion along the various axes of different crystals 
is investigated very interesting results are obtained. In the hexago- 
nal system, for optically negative crystals the expansion along the 
axis is always greater than that along an axis at right angles 
to it ; while for optically positive crystals the reverse Is the case. 
Thus, for Tcelaud spar we have expansion parallel to the axis and 
contraction perpendicular to it. The contraction is always much less 
than the expansion so that the volume coefficient remains positive. 

EXPANSION OF LIQUIDS 

15. In case of liquids we have to consider only the cubical ex- 

am be expressed as a fu 
anperature ; thus 

F= P (14-«it |-«,-' : — • )» - • (12) 
or ap , ,, 

r«(j *t), ■ ■ ■ ■ (is) 

ere « is called the mean coefficient of expansion between 
i t*G. Thus if a mass M of the liquid occupies the volumes F, J ',, 
at P and 0°C, the densities p, Po of the liquid at the respective 
temperatures arc p =Mf V, p*= M/V . Using (13) we get the relation 

" £-.l+«fc / . , . (H) 
P ** 

The expansion of liquids is much greater than that of solids, yet 
il is more difficult to measure, for it is complicated by the expansion 
of the containing vessel. The expansion _ observed is called the 
apparent expansion and is a combination of the two effects, 
'.. expansion of the liquid and of the containing vessel It can 
be shown (see sec. 17) that the coefficient of absolute expansion of 
the liquid is approximately equal to the sum of the coefficients of 
expansion of the containing vessel and the coefficient of apparent 
expansion of the liquid. Thus the former can be determined if 
latter two quantities are known, 

* Further sec Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 876. 






WEIGHT THERMOMETER METHOD 



167 



There are three well known methods for determining the appa- 
rent or relative expansion :— « B jJ,»TW«if! 

(i) The Volume Thermometer Method, hi) the Weight Thermo 
meter Method,' and (Hi) the Hydrostatic Method. 

16. The DSatometer or Volume Thermometer Method- The ^ 

S* divMon of Ltd ! then volumes of the liquid at the two 
temperatures are ^ + ^ ^ (F# + ^ (I + ^. 

y being the expansion of the containing vessel. The volume at 

t°C. is also equal to 

(F« + *ift) (! + «*>' 
where « is the coefficient of absolute expansion of the liquid, Equa- 

1 ^ehave^ ^ < 1 + -ftsB <r8+% ^ (l + rf> 

Knowing y the true coefficient « is calculated, or if 7 is not known, 
the relative expansion «-■ y can be evaluated . 

17. The Weight Thermometer Method.-* more accurate method^ 
deluding upon the determination of weight and not of volume 
iskmishld by the pyknometer or the weight ttarai^cki. Thtse 
are vessels so constructed as to take a definite volume of t liquid. 
The weight thermometer is of the shape shown 
in Fig. 8, and is made of glass or fused silica. 
It is first weighed and then completely filled with 
the liquid by alternate heating and cooling with 
the open end dipping in a cup of the liquid. 

The experiment consists in weighing the 
thermometer filled with the liquid at two tem- 
peratures. Let it.-!, Wi represent the weights 
of the liquid filling the thermometer at tern- 
peratures t x and t 2 respectively. If Vj, F 3 are 
the volumes of the vessel at the two tempera- 
tures \md p v p 8 the corresponding densities el 
die liquid, then 

w x = F 1PI , u> 2 — Vzp* • (15) 
if a, y denote the expansion coefficient 
[quid and the vessel respectively, 




Fig. 8.— The W i£ 

Thermometer. 






; +vf* 



l+rh 

w 2 l-hyr 2 



Pj__l+nh 
Pi l+a/2 

1-1 oV 



. . - ' . (16) 

—X) ih—h) approx. (17) 

The apparent expansion a can be obtained from (16) or (17) if 



168 



THERMAL EXPANSION 



GH IP, 



the expansion of glass is disregarded i.e., y is put zero. We then 
obtain from ; 



or 



-S — 1 -}- a{t « — U) app rox. 

a=* — - L_ approx. 



(IB) 



Equation (16) can be breated rigorously. Assuming t 2 to re£er 
to 0°G, and dropping the suffix I, equation (16) yie 

w _ 1 ! , 
BL 1 ! at ' 



or 



a = 

wt w 



(19) 



It is thu^ seen that, rij ly speaking, the true expansion coeffi- 

cient is a little more than the sum of the apparent coefficient and 
the expansion coefficient of glass, though the difference is almost 
negligible, and for all practical purposes we can assume a = a-$-y> 
We could treat equation (10) im rally when the result will 

be more complicated than (20) . Knowing y the absolute expansion 

in be calculated. 

The therm ometi nployed to find the cubical expan- 

ig the specimen inside the thermo- 

:r.* 

18. Hydrostatic Method (Matthiessen's method), — This consists ill 
finding the apparent weight of a solid when immersed in the liquid 

oectivcly. The loss in weight ol: the 
solid is by Archimedes' principle equal to the weight of a volume of 
the liq lal to that of the solid; denote this quantity by w. Then 

where V Xt V% denote the volumes of the solid at the two temperatures 

t lt f 2 respectively. Then 

v i _ 1 +y<i Pi_l±al s 

K " ! l ; 

and proceeding as before, 

i ~M? a 



P 9 1 4- <*V 



a—y = 



approx. 



(21) 



An equation analogous to (19) can also be deduced. 

19. Absolute Expansion of Liquids. — As already mentioned, the 
three foregoing methods may be employed to find the absolute 
expansion of a liquid provided the cubical expansion of the contain- 
ing vessel [or of the immersed solid in § 18] be known. One way 



* Sec Glazebrook, A Dictionary of Applied Physics, Vol, 1, p 



vn. 



HVOROSTAT1C BALANCE METHOD 



169 




Balance Method. 



of finding the latter is by calculating it : linear expansion. This 

is, however, open to objection for Che linear expansion is deter mined 
from bars of the material and it cannot be assumed a priori that the 
physical i es of the material do not change when it is an- 

nealed and worked into a vessel of some shape. For this reason it 
is best to select vessels of fused silica for which the volume coeffi- 
cient is extremely small (about 0.0000015 per °C.) . 

20. Hydrostatic Balance Method*— There 

thod of determining the ahsolute expan- 

i of a liquid which was first given by 
Dulong and Petit. It depends on the hydro- 
static balancing of two liquid columns at 
different temperatures. Dulong and Petit 
employed a simple U-tube for the purpose. 
•Regnault brought the upper ends of the tube 
close together, an improvement which made 
it easier to observe the difference in height of 
the two columns. The diagram (Fig. 9) 
scives to illustrate the principle ol the 
method. A glass or metal tube, bent as 
shown in the figure, contains mercury. The 
vertical columns AB, CD, C'D r are sur- 
rounded by melting ice and are thereby 
maintained at (PC, while the column A'B' is surrounded by an oil 

maintained at any temperature i°G. Suppose that AA' is 
zontaL Let H, H r t h, h* denote the heights of mercury in the various 
columns as shown and p, p Q the densities of mercury at t a C, and 0°C. 
Then since the pressures at D and D' are equal, we have by equating 
the two expressions Cor the hydrostatic pressure at A, 

h?p t +H , t> = Hpo+hfr. .... (22) 
But 

where c is the coefficient of absolute expansion of mercury. Hence 

j^+AWf+A, .... (23) 

whence c can be calculated. 

If the columns H, h, h f are not at 0°C. but at temperatures 
, f 3 respectively we shall gee 

*: + . A ' . _ .jl h t- . . ( , 

1+rf ^ l+c^ ' " 1 + ^f, r H-c a '« ' 

where the quantities c-,, c>, c ? , denote the mean coefficients of expan- 
sion between the different ranges. These can be determined by 
having the temperature of A'B' to be £ 1? t%, ( s successively, 
the height of A' above A is /i la a corresponding term can be adr 
the right-hand side. 

Regnault's observations, though carried out with greal 
must be corrected for various sources of error and hence cannot vi: Id 



170 



1H BRM A I, EXPANSION 



CHAP- 



n 

rttiti eta ij 



CH 



Fm. 10.— Arrangement of 

Callendar and Moss's 

apparatus. 



results of high accuracy, Callendar and Moss repeated the experi- 
ments aiming at a high degree of accuracy. Instead of a single 
of hot and cold columns 1.5 m. long employed by Regnault, they 
wsed six pairs of hot and cold columns each 2 m. long and connected 

in series as shown diagrammatically in 
Fig. 10, The hot and cold columns are 
marked H and C respectively. The differ- 
ence in height of the first and the last 
column (viz., &b) is six times that due to 
a single pair. 

In the actual apparatus ef, gk , . . were 
doubled back so that all the columns marked 
C were one behind the other, and similar 
was die case with H columns. All the H 
columns were placed in one limb of a rect- 
angle and all the C columns in one limb of 
another rectangle, while the o titer limbs of 
these rectangles contained electrically heated oil and ice-cooled baths 
respectively. These were kept circulating by means of an electric 
motor and their temperatures were determined by a long 'bulb' re 
tan cc LliiiriTometer, Experiments were performed in the range to 
300°G. and an accuracy of 1 in 10,000 was aimed at, 

21. Results for Mercury.— Their values for mercury are, howe 

ifEerent from the mean of earlier investigators such a* 

; . Harlow hi made accurate dctermi- 

i the help of a weight thermometer of silica and aimed at 

of 1 in 18,000. The concordance o£ results with bulbs of 

,ed that as quite isotropic. Th ■ ient 

of e i in the region to 100°C. is according to Callendar and 

C. 

The following numerical examples show how the expansion of 
rcury Is taken into account in (a) the correction of the barometric 
,ture, (b) the correction for the emergent colum 
of a mercury thermometer and (c) the compensation of the mei m 
ttm. 

Example 1 . — A barometer having a steel scale reads 750.0 mm. on 
a day when the temperature is 20°C. If the scale is correctly 'gradual 
at o G., find die true pressure, given that the coefficient of linear ex- 
pansion of steel = 12 X 10~* C C~\ and coefficient of expansion (abso- 
lute) of mercury = 182X IO - ' 5 per °G 

The length of scale at 20"C = 75.00 (I + 20 X 1£ X MH") cm. 
Density of mercury at 20°C ^= p f (1 + 20 X 1S2 X 1{) ~ 6 ) 
where p Es density of mercury at D C. 
75.00(1 4- 20 k 12 x 10^)^ 



ressurc — 



dyncs/cm a * 



(1+20*182x10^ 
=75.00(1-0.0034) ^=74.745 Ptt g dyncs/cm.* 



VII.] 



EXPANSION OF WATER 



171 



Example 2. — A mercury thermometer, immersed up to S0°C mi 
in a hot liquid reads 230*0. If the exposed stem has an a 
temperature of 50°C, calculate the true temperature, given that 
mean coefficient of expansion of mercury is 182 >< 10-* per n C, and 
coefficient of linear expansion of glass = 8 X W- 6 D C -1 - 

Coefficient of apparent expansion of mercury 
= (182 - 3 X 8)10-* = 158 X 1CM °C^ 

Hence, the exposed stem, if at the true temperature t*fcl., would 
occupy a length (230-30) [1 + (£-50) 158 X 10"*]. 

/. ( - 230 = (230 - 50) (i - 50) 158 X H)-*, 

when * = 235 U C. 

Example 3. — F In a mercury pendulum a steel rod of length I cm. 
D C supports a glass cistern containing mercury, find the height 
to which mercury should be filled up in the cistern for perfect compen- 
sation of the pendulum, given that the linear coefficient of expansion 
of steel a '= 12 X 10- G , linear coefficient of expansion of glass g = 8,5 
X 10"*, cubical coefficient of expansion of mercury m = 1.82 >( 10-' 

I .et h be the required height of mercury in the cistern at C 

V the volume of that men nd A the cross-sectional area of the 

at n C. Due to the rise of temperature to t c 'C, the 
volume ' ' 7 (1 -J- mt) , the cross-sectional area of 

the cistern increases to A (1 -2gf) and therefore the height of mer- 
cury increases to 

Since the centre of gi \\ ic- of mercury at 0°C is at. a height ft/2 
from the bottom, it will rise to (ft/2)(l -\-mt-2gt) at t a C, the in- 
crease being (h/2) (mt — 2gl). The increase in the length of the steel 
red at t Q C is kit. For perfect compensation these two changes must 
be equal. Hence 

imt-2gt) = lot. 



Oi 



h =- 



2 a 



2-2 



•'; 



= _2xl2xl0^_ 

(182-17) xlCr-* ' tM * J <" 



22. Expansion of Wafe*. — Tt is well-known that the expansion 
of water is anomalous in the region to 4°C, Several workers such as 
Hope, Despretz, Ma tth lessen, Joule and Play fair and others, measured 
this expansion carefully. A constant volume dilatometer (sec. 16) 
may be employed for this purpose. If the dilatometer is made of 
ordinary glass some mercury is initially put in it to compensate for the 
expansion of glass. Since the expansion of mercury is 0.000182 and 
of glass 0.0000255, a volume of mercury equal to due-seventh of the 
volume of the dilatometer will be required for compensation. These 
experiments show that when water at 0°C is heated it goes on con- 
tracting as long- as the temperature is below 4°C. Above 4°(] it 



172 



THERMAL EXPANSION 



[CHAP. 



expands on heating. Accurate experiments by Joule and Playfair 
show that this temperature of maximum density is 3,95 °C. This 
anomalous behaviour is usually explained on the assumption that, there 
exist three types of molecules H 2 0, (H 2 0) 2 , (HgO) 8 , which have 
different specific volumes and are mixed in different proportions at 
different temperatures. The total volume occupied is assumed to be 
the sum of the specific volumes, though there seems to be little 
justification for such an assumption. 

PRACTICAL APPLICATIONS OF EXPANSION 

23. The expansion of solids and liquids is of great importance in 
daily life and its consequences have often to be borne in mind 
carefully. The student will be familiar with most of these from his 
elementary studies. Thus it is well-known that allowance must be 
made for expansion in the laying of railway lines and the erecting of 
.steel bridges, the contraction of metal tyres on cartwheels etc. The 
expansion of tine steel scale must be taken into account in reading 
the eter (sec. 2.U Example 1). Similarly the expansion oi 

mercury must be found out for obtaining the true temperature fro 
a mercury thermometer (sec. 21, Example 2). Expansion in optical 
and electrical apparatus also causes difl i. The mirrors of 

reflecting telesc ich are very accurately figured must be pro- 

tected from distortion of the ce by expansion of dij . por- 

different temperatures. The i tee of a 

coi; on its diirif.r changes with expan- 

sion con. 

In dealing with glassware it is particularly Important 

•i and contraction as the glass is lik 
to ;:• I to sudden changes of temperature. Thii 

oor condi heal and the difference in tem- 

iure between the different parts causes unequal expansion. 
;e is avoided by either choosing thin glassware or choosing 
: ; of small co-efficient of expansion and high thermal con I [vity. 
Fused si sels ran he heated to white heat and plunged safely 

into cold water, while pyrex glass can be thrust into a blowpipe flame 
suddenly without risk of fracture. Platinum wires can be sealed into 
ordinary glass (lead glass) without any risk of a crack developing 
because platinum and glass have practically the same coefficient of 
expansion but a copper wire would cause cracks to develop due to un- 
equal expansion. 

24, Compensation of Clocks and Watches. — For our present pur- 
pose the pendulum of a clock may be treated as a "simp i . du- 
rum" with a bob of negligible dimensions suspended at the end of a 
wire o£ negligible mass. "The time kept hv the clock depends upon the 
time of oscillation of its pendulum which, in the case of a simple 
pendulum, varies as the square root of the length or the pendulum. 
The length of the simple pendulum is the distance between the point 
support and the centre of gravity of the bob. Thus if this length 



COMPENSATION OF CLOCKS AND WATCHES 



173 



vn.j 

increases in summer due to expansion, the time o£ oscillation will 
increase and the clock will lose time. If the length decreases due to 
fall of temperature, the pendulum swings faster and the clock gains. 
Therefore unless the pendulum is compensated against the effect of 
expansion the clock will gain in winter and lose in summer. This 
compensation is brought about by the use of two expansible materials 
so arranged that the expansion of one is compensated by the expan- 
sion of the other. 

In the gridiron pendulum alternate rods of steel and brass are 
connected as shown h\ Fig, 11 so that the steel 
rods expand only downwards, and the brass rods 
only upwards. It is so arranged that 7,cu=ta*2 
where l x and l 2 are the total lengths of the steel and 
brass rods respectively, and «i, og the respective co* 
efficients of linear expansion. Under these condi- 
tions the effective length of the pendulum remains 
constant at all temperatures and the pendulum is 
compensated. It will be seen that J a /1 2 == W a i ~ 
3/2 a pproxi m ate ly, 

In the mercury pendulum a long steel rod 
carries, in place o the ordinary bob, a frame fitted 
with two glass cylinders containing ary. Com- 

pensation is obtained by the expansion of the rod 
downwards and the expansion of mercury upwards, 
(See sec. 21, Example o) . 




Fig, II, — Gridiron 
pendulum 



In watches the rate of movement is governed by 
the oscillation of a small flywheel called "the balance 
wheel which is itself controlled by a hairspring. The 
time of oscillation of the balance wheel depends 
upon the stiffness of the hair-spring and the 
moment of inertia oE the wheel about its axis of 
oscillation. The moment of inertia depends on the diameter of the 
wheel and is mainly contributed by the three small weights (Fig. 12) 
screwed on the rim of the wheel. A rise in tempera- 
ture weakens the hair-spring and increases the dia- 
meter of the balance wheel. Both these changes 
cause the oscillations to become slower, the former 
effect being more important. To effect com] 
tion, the rim of the balance wheel is made in 
segments of a bimetallic strip with brass on the out- 
side and steel on the inside, and weighting the rim 
with three weights as shown in the figure. Due to 
the larger expansion of brass the end of each segment 
curls inwards when the temperature rises thereby 
reducing the moment of inertia sufficiently to compensate both for 
radial expansion and weakening of the hair-spring. The precision 
watches such as chronometers are usually provided with cor i 
balance wheels. Invar steel is now frequently used in the manufacture 




Fig. 12.- 
Compensated 
balance-wheel. 



174 



THOtMAL EXPANSION 



[CHAP. 



of pendulums and balance wheels on account of its very small coeffi- 
cient of expansion,. 

25. Thermostats. — The property of expansion is often utilised for 



* 



tnr 




13.— Toluene Thermostat 



constructing thermostats. In these the 
temperature of any substance can be 

kepi constant for a long time. For 
temperatures upto lGG a C s a toluene 
thermostat may be used but 'for tempera- 
tures above JOCC, a bimetallic thermo- 
regulator is generally used. These will, 
be described. ' 

Tolu-ene Thermostat, — A toluene 
thermostat is shown in Fig. 13. In the 
bulbs T there is toluene, alcohol or 
some other liquid having a large co- 
efficient of expansion. These bulbs 
are immersed in the bath whose 
temperature is required to be maintained constant. In case the tem- 
perature of the baLh increases the toluene expands and forces the 
mercury (shown black in the figure) up the tube in C and thereby 
ises the opening leading from A to B. The bath is heated by a 
burner and the gas ' his hunter is supplied through A and this 

opening. Thus, on account of the expansion the gas supply is 

1 the temperature falls. Tf the temperature of the bath 

falls too much, toluene contracts, the opening- is increased and 

By adjusting the amount of mercury in K. the 

ire can be kept constant at any desired value. In case of 

trical heating electrical contact can be arranged above 

the mercury lev:: ft.chmg off the heating current. 

illic Therma-Regulator. — The Dekhotinsky bimetallic thcr- 
ulator is shown in Fig. 14. It. consists of a compound strip R. 
which is made by welding to- 
gether brass and invar steel. 
The scrip is wound into a small 
helix and is connected by 
means of the rod P to the con- 
tact maker C which makes or 
breaks the electrical circuit of 
the heater H. When the tem- 
perature rises the expansion of 
the brass causes the helix to 
unwind and thereby the con- MAINS 

tact at C is opened thus break- „...„._,,,,.„. ^ _ . . 

. . £ , • ., ,,,.,- Fig. 14. — I be Bimetallic Thermo-Rcgulator. 
:c electrical circuit. With 

c- the helix contracts and thus contact is made at C 
thereby completing the electrical circuit. With this apparatus tem- 
peratures up to 3(K)°C can be maintained to within ±1*C. for a 
long time. 




VII. I 



EXPANSION OF GASES 



175 



(27) 



EXPANSION OF GASES 

26* Expansion of Gases*— The expansion of gases forms the very 
basis of the system of thermometry and the perfect gas scale discussed 
in Chap. I. As stated there the results are best expressed in the form 
of Charles' law which holds very approximately for the so-called 
permanent gases in nature. Here we shall describe the experimental 
methods of determining the coefficient o£ expansion. 

In the case of gases it is necessary to distinguish be L ween 
coefficients of expansion : (1) the volume coefficient of expansion « at 
constant, pressure, and (2) the pressure coefficient of expansion $ ai 
constant volume. The volume coefficient of expansion is denned 
as the increase in volume of unit volume at a C for each centigrade 
: ■■:.?. rise of temperature at constant pressure. Thus 

<^ = K V J L . (25) 

where V and F n denote the volumes of a fixed mass of gas at t° and 
II : C, Or 

F=F*tl + «9 (26) 

If the volumes at temperatures t\ and f 2 a* 6 Vi and F 2 respectively* 
get, with the help of (26). 

«~ Eta .... 

We can thus determine a by measuring the volume of a fixed mass 
of gas at two temperatures. 

Similarly the pressure coefficient of expansion of a gas is defined 
as the increase in pressure, expressed as a fraction of the pressure 
at 0°C. for one centigrade degree rise of temperature when a fb 
mass of the gas is heated at constant volume. Thus if p and p^ be 
the pressures at t Q and 0*C we have 

from which relations analogous to (26) and (27) can be deduced. 

27. Experimental determination of the Volume Coefficient of Expan- 
sion. — Gay-Lussac was among the earliest to measure the volume 
'licient accurately. Resmault used an improved form of apparatus 
and corrected his results for various sources of error. Fig. 11 shows 
a laboratory arrangement for determining the volume coefficient and 
employs Regnault's technique in a simplified form. 

e bulb A is connected by a narrow glass tube to a calibrated 
limb B of a mercury manometer whose other limb C can be moved 
up and down for adjusth ry level in. B. The tap T enables 

the quantity of gas in A to be adjusted. First the bulb A is put in 
j cold water bath at t t °, and after it has acquired the temperature 
of the bath, the tube C is adjusted until the mercury stands at the 
same level in both arms, and the mercury level in B noted. Then the 



*-- 






176 



THERMAL EXPANSION 



[CHAP* 



is heated, when the enclosed air expands pushing the mercury 
down in B and up in C, The bath is maintained at a certain tempe- 
rature and the tube C lowered to bring 
the mercury level at the same height 
in B and C, and Uie volume of gas in 
B read from the graduations. The 
process is repeated for every 20° rise 
of temperature up to I00 c C, and the 
observed readings are utilised for cal- 
culating a from (27) . 

The results are hest treated by 
plotting the observed volumes against 
temperature on a graph* It is found 
that all the points lie on a straight line 
showing that equal changes in tem- 
perature lead to equal changes in 
volume at constant pressure. This is 
Charles' law which may be formally 
stated : for a fixed mass of gas heated 
at constant pressure, the volume in- 
creases by a constant fraction of the 
■me at 0°C for each cem . 
ree rise in temperature. This also 
follows from the result that a comes 
out to be the same whatever values of 
2 are utilised in (27), Results 
further show that a = 1/273 (nearly! 




...'.. 



Hed permanent gases 

For accurate work ecu 

)e applied for the following 
sou, : — (1) the gas in the 

ow tube and the manometer is at 
a different i emperature From the bath, 
on of the glass bulb 
with rise of tern re. Regnault 

appli erections for these and 

found that all real gases showed small 
departures from uniform expansion 
and that the coefficient of expansion 

red slightly from one gas to 
another. 

28. Experimental determination 
of the Pressure Coefficient of Expan- 
sion.— -The pressure coefficient can be 

y determined in the laboratory 
with tiie help of an apparatus 
known as Tory's apparatus. It 
consists of a glass bulb A, of about 
1 C.C. capacity, which is filled with 
dry air and is connected by a glass 




Fig\ 16. — Joly's apparatus 
for determining pressure coefficient. 



VIL] 



PRESSURE COEFFICIENT OF EXPANSION 



177 



capillary tube to a mercury manometer mounted on a stand (Fig, 16) . 
A fixed' reference mark X Ls made on the tube B near the top by 
means of a file, and the mercury level is always brought to this mark 
by adjusting C before any reading is taken. This ensures that the 
volume of the enclosed gas is kept constant. A metre scale S is fixed 
to the vertical stand to read the difference ft in the levels of mercury 
in the two tubes B and C. The bulb A is immersed in a water bath 
which is well stirred, and temp.Liai.Lin:- are read with a mercury 
thermometer. 

First the experiment is done with cold water in the bath and 
die difference h m the levels of mercury in the two columns noted. 
If the barometric height is H, the pressure of the gas is H ±h depend- 
; ion whether the level in C is higher or lower than that in B. 
The bath is dien heated through about 20°, heating stopped, and 
the bath well stirred. The mercury is again brought to the reference 
mark and the level watched carefully. When the level becomes steady 
at die reference mark, the reading- in C is noted. Heating is then 



resumed and readings are taken in this way at intervals of 2Q\ 



The 



j>- 



(29) 



coefficient of expansion /3 is then calculated from the relation 

Pz -Pi 
P±h—p.: : 

For accurate work various corrections are necessary. The most 
difficult to estimate is the "dead space" correction (p. 9) since the 
exact temperature of the gas in the capillary tube is not known. 
The expansion of the bulb introduces an error in /S of the order of 
1% ; this can be satisfactorily corrected for by adding the coefficient oi 
cubical expansion of glass to the observed value of p. Experiments 
shown that p is fairly dose to 1/273 for all the permanent 
.•.•; which means that for a fixed mass of any gas heated at constant 
volume, the pressure increases by 1/273 of the pressure at 0°C for 
each centigrade degree rise in temperature. Accurate experiments 
however show thai "this expansion is neither uniform for a gas, nor 
is it exactly the same for all gases. 

As mentioned in §7, Chap. L these observed deviations of a and j3 
from the correct value of 1/273.16 are really due to the deviation of 
actual gases from Boyle's law. Let a fixed mass of perfect 
which by definition obeys Boyle's law, have the pressure ad 

ioe t-'o at 0°C, When it is heated to t°C, the product of pre 
and volume can be written as jfy>#o (1 + at) or ptft/tt (1 -f- /?r) depending 
on whether the pressure is kept constant- or the volume is kept cons- 
Since these products must be equal by Boyle's law, we get a — /3 
lor a perfect gas. The expansion coefficient is experimentally found 
to be 0.0036608 for all gases provided they are reduced to the state of 
a perfect gas (p— >Q) . "The equality of « for all gases is really a 
consequence of the kinetic theory. 

Book Recommended 

A Dictionary of Applied Physics, Vol 1,. 



pp. 



1. Glazcbrook 
872—900. 

12 



CHAPTER VII! 
CONDUCTION OF HEAT 

1. Methods of Kent Propagation. — When a bar of metal is heated 
at one end and held at the other by the hand we ■ nsation of 
heat Heat travels from the hot end along the bur and produces this 
sensation in the hand. The power of transmitth manner 
is possessed by all substances to a varying degree and this phenomenon 

tiled Conduction oj Heat, In the process of Co m, heat in 

transferred by the actual motion of heated parricli s of matter whether 
liquid or gaseous. Tins is best illustrated by placing gently some 
ils of potassium permanganate ;il the bottom of a beaker conn 
f ainhig water and heating it. Heated water rises up and curls down 
forming a closed path which is rendered visible by the red colour 
imparted to the water. In conduction, heat is transferred by 'contact* 
and there is no apparent transfer or matter. 

In both the above processes the intervening medium takes an 

active part in heat propagation, I mi in addition to these there is 

anoiJi: i i . ii ■■ the intervening medium takes no part, il 

id near a coal furnace, we feel the sensation of heat. If we 

■ the iuzi| we fee] w\ are the source of heat is the 

e sun, and In the latter case it is at an enormous distance 
io mate] ial medium b< tween us am 

M.. This phenomenon 

is call r'hc processes of conduction and convection arc 

./.,- due to the action of the intervening medium while 

radiatii with the enormous velocit; Eit. In this chaptei 

aduction of heat. 

2. Conductivity of Different Kinds of Matter. — Common observa 
show that different substances vary enormously in their con- 
ducting power. A glass rod can be melted in a flame by holding ii at 

[nt two or three inches away from the Game, while a copper rod 
under similar conditions becomes too hot to Couch. Copper is thus 
i better conductor of heat than glass. Metals, in general, are good 
conductor-, ol heal ; glaj iod and other non-metals are bad con- 
ductor*, Hold inside a flame two blocks, one of wood and the othev 
of copper, each covered with paper. The paper covering the copper 
1 -I'M i: i , not burnt, for heat is rapidly conducted away h r and 

mperature does not rise it .nition point. The' paper cover- 

he wood is burnt, Similarly, wat< be tied in a cup of 

paper for the; heat, is taken up by water (convection) , 

Liquids, in general, are worse conductors of heat than solids and 
are the m \ very simple experiment shows thai water is a 

bad conductor of heat. Take some water in a test-tube and sink 
into it a piece of ice by weighting ii. The water at the top can be 
boiled by Ilea ting it locally while the ice at the bottom docs not even 



VIII.] 



DEFINITION OF CONDUCTIVITY 



179 






melt. Air and gases in general, are even worse conductors than water. 
Woolen clothing protects us from cold on account 
of the fact that it contains air in the interstices which 
renders ir a very bad conductor of heat, 

The good conductivity of metals is utilised in the 
construction of the Davy safety lamp. The flame is 
enclosed in an iron gauze chamber (Fig, 1) and can 
be taken down a mine. Any combustible gas, if 
nes in contact with the naked flame and 
burns inSide the chamber. But the iron gauze con- 
ducts awa; heat so quickly that the temperature at 
any point of it does not rise Co the ignition point and 
the outer gas does not ignite . 

3. Definition of Conductivity.— The first, to give 
a precise definition of conductivity was Fourier who 
in his memorable Theorie Analytique de la ChaJeur 
(1822) treated the subject of heat conduction in a 
masterly way and placed it on firm mathematical 
basis. We shall first discuss the (low of heal inside a 
bar heated at one end. Consider a thin wall of the material with 
u irallel Fa< i . such that heat Hows in a direction perpendicular to the 
Lftl Ou 0fl be the temperatures of the two faces, I the thickness 
rea of each lace, then it can be shown experi- 
mentally that the amount of heat Q flowing through the wall in a 
time f, when the temperature at every point of tHe bar is steady, 
is directly proportional to (t) Qi-6 2 > 00 tn e area A of the surface, 
{Hi) the time t and (iv) inversely proportional to I, that is. 




Fig; 1, — Davy 
Safety Lamp. 



Q,= KA 



l 



(1) 



The coefficient K is a quantity depending upon the nature of 1 1 1 ; 
iubstance and is called its thermal conductivity. From this relation 
conductivity may be dqfined as the quantity of heat flowing per 
tnd through a unit area of plate of unit thickness, when the 
difference of temperature between the faces is unity. In analogy 
with the electrical resistance, the inverse of K may be called the 
distance of a unit cube, 
\Yi-,v imagine the thickness of the plate to be diminished indefi- 
nitely. The limitino Milne of * , a or --,- l is — and denotes 

/ / dx 

the temperature gradient at any point. The minus sign has been 

used befi n ", because the symbol d always stands for the increase. 

Hence, the quantity of heat flowing in the positive direction of x 
in time df across the isothermal surface of area A at any point x is 
given by 



d--KA^di. 



(2) 






180 



CONDUCTION OF HEAT 



CHAP. 



This equation is of fundamental importance in the theory of heat 
conduction. The Links in which conductivity is measured in the CCS. 
system are the calorie per second per square centimetre of area lor a 
temperature gradient of 1 G C. per cm. [cal cm, -1 seer* 1 ''C- 1 ]* 

CONDUCTIVITY OF METALS 

4. We shall now give the various methods of deter mining the 
thermal conductivity of metak. Methods I to III employ stationary 
heat-flow while in IV a stationary periodic flow of heat is used. 

I. Conductivity from Calorimetric Measurement 

5. The definition of conductivity from equation (1) provides a 
simple method of determining the conductivity of a substance. We 
need have only a slab of the material of known cross-section heated 
at one end, and measure the amount of heat that flows out at the 
other end in a known time, as well as the temperature of the two 
faces. Titus all the other quantities except K in equation (1) are 
known, and hence K can be evaluated. 

The method though simple presents considerable experlmem 
difficulties. It is difficult to measure accurately the temperature of 
the two faces of a slab of metal. This is best achieved by keeping 
.embedded in the surface the junction oF a thermocouple, the use of 
mercury thermometer or resistance thermometer being inconvenient 
or impossible. Some early experimenters used steam to heat the slab 
at one end and ice or n cool it at tfoe other end, and assumed 

ture of the end rare, of the slab was that of steam 
and water res But if proper precautions be not taken, the 

od .sometimes gives absurd results This is on account of the 
fact that a thin film of fluid always at rest, is formed in contact with 
the si and this has a large temperature gradient. Hence it is 

essential that we should observe the temperature inside the slab itself 
by means of thermometers. There are many apparatus based on this 
method, and one due to Searle is described below. 

Fig. 2 shows the apparatus diagrammatically. One end B of 
the rod AB i& enclosed in the steam-chamber S while the other end 

A projects into another chamber C through which cold water circulates 
as indicated, the temperatures at entry and exit being T a and T 4 res- 
pectively. Temperatures at two points along the bar are measured 
by thermometers T x and T £ ; let these be T x and 7V The whole rod 
is wrapped round with some non-conducting material Tike worn 1 
etc. In the steady state if m grams of water How past the con; 
of the bar per second, the heat conducted by the bar per second is 

T —T 

1 -r— S- , where d is the distance 



?n(Ti—T s ) and this equals &K 
thermometers T lt T 



Thus, the conductivity K is 



between tin: 
determined, 

In the foregoing experiment some heat was lost bv radiation 
from the sides of the bar. This is a source of error and is very 



vni. 



CONDUCTIVITY FROM CALORIMETRIC MEASUREMENT 



181 



easily eliminated if the bar be surrounded with some material at 
the same temperature as the adjacent portions of the bar. There 
will be no How of heat perpendicular to the length of the bar as no 




Fig. 2. — Searle's appara 

temperature gradient exists in that direction. The surrounding 
material is called the 'guard-ring'. 

Beige l utilised this guard-ring method for determining the 
conductivity of various substances such as copper, iron., brass and 
mercury. A vertical cylindrical column of mercury is surrounded 
by an annular ring of mercury. In both the upper surface of mercury 
is heated by steam while the lower surface rests on a metal plate 
Thus the temperature gradient as well as the 
temperatures at the same level in the experimental column and the 
'surrounding annular ring are identical! Under these conditions 
there can be no lateral flow of heat and the annular ring serves as 
a 'guard-ring'. The lower end of the experimental column projects 
into die bulb of a Bunsen ice-calorimeter and the heat flowing out 
at die lower end is found from the indications of the mercury 
thread. The difference of temperature between several points along 
the column is determined by four differential thermo-couples. The 
conductivity K can be calculated from formula (1) . 

This method has been adopted by a number of workers, notably 
by Lees, Donaldson, Honda and Simidu, and others. Lees used 
the rod method for measuring the con- 
ductivity of many metals throughout the K-f 
temperature range 18°C to - 170°C The 
imental rod, about 7 cm. long and 
0.58 cm. in diameter, was heated electri- 
cally at the upper end, its lower end being J£- 
fixed to the base of a hollow, closed, 
cylindrical shell, of copper which complete- 
ly surrounded the rod. The outer cylinder 
was suspended inside a Dewar flask and immersed in liquid air or 
heated electrically and thus the desired temperature of experiment 



-i jl 
p. 



Fig. 3. — The two-plate * 
system. 



182 



CONDUCTION OF HF.AT 



CI3AP. 



was attained. Knowing the electrical energy spent in the heater 
wire and the temperature at two points on the rod by platinum 
resistance thermometers, the thermal conductivity can be calculated. 
Some corrections are however necessary which are difficult to evaluate 
accurately. 

Honda and Simidu employed the two plate method shown in 
Fig'. 3. P|, P 2 are two exactly similar plates made of the experimen- 
tal material. Between these is symmetrically placed the electrical 
heater H in which the heat Q is generated. ' For perfect symmetry 
two cold plates, K, K maintained at the same temperature are placed 
at the other ends of these plates. Thus the amount of heat flowing- 
through each plate is Q/2, 

IL Conductivity from Temperature Measurement — 
Indirect method 

6. Rectilinear Flow of Heal, Mathematical Investigation,— 

Consider a metal rod heated at. one end. 
the isothermal surfaces being parallel 
planes perpendicular to the length of 
tlie rod, and let the axis o[ x be normal 
to ihe.se planes. At a distance x from 
the hot end (Fig. I) let Q be the tempera- 

ture and — the temperature gradient, 

t of thickness v at this point. The amount of 

di) 

(from equa- 
tion 2), Qz the heat which leaves the layer at the face x-\-dx is 

1 










6+dd 


Qt— 


Lq ' 


Heale 


r o~ - 

r 


X 


3 a 



k— Flow of lieat in a rod. 



heat Q, whid . the lavei per second is -K. 

dx 



—KA 



ii' 






since $ -; is the temperature of that face. 



dx 



Hence the gain 



of heat by the layer is equal to 

Now before the steady state is reached this amount of heat raises 
the temperature of the layer of thickness dx. Let p be die density 
and c tiie specific heat of the material per unit mass, and let the 

rate of rise of temperature be denoted by - . The mass of the layer 

Idx. Hence, neglecting the heat lost by radiation from the 
surface, we get 



KA—r-a dx = pAdx. C 



di 



JC dW_ _ d*9 
pc dx* " dx* 



- (6) 



via.] 



rectilinear elow of heat 



183 



where h =- K/pc Thus h is equal to the thermal conductivity divided 
by the thermal capacity per unit volume. This constant k has been 
called thermal difjttsivity by Kelvin and tkermonwtric conductivity by- 
Maxwell but the former term is more commonly used. It will be 
seen that the thermal dilfusivity h represents the change of tem- 
perature produced in unit volume of the substance by the quantity of 
would flow in unit time through unit area under unit 
temperature gradient. Thus for calculations of the rate of rise of 
temperature the constant k is of greater importance than K as equa- 
tion (0) shows (as the rate depends not only on K but also on pe) t 
but in the steady state the rate of flow of heat depends only on K 
and not on p c (see equation 8). 

Tf, in addition, the sides of the bar are allowed to lose heat 
by radiation, this must be taken into account. If E is the emissive 
power of the surface, p its perimeter and $ the excess* of the tempe- 
rature of the surface over that of the surroundings, this radiation 
loss 3 , assuming- Newton's Law of Cooling to hold true, is equal 
to Epfi dx. Hence we should rewrite (5) as 

d& 



Or 



where 



KAi-.dx^p Adx.c Tl 

ax at- 



m 






E P 



pAc' 



'litis is the standard Fourier equation for one dimensional 
flow of heat and any problem in thermal conduction along a rod 
consist-: simply in the solution of this differential equation. 

Steady State — A state is said to be steady when the tempera- 
ture at every point of the rod is stationary, Le. 3 -j - — 0. We have 
then 



where 



ST- 5 * - ««l 

m' = !i = Kp . 

h AK 



(*) 



If the radiation losses from the sides can be neglected, this 
equation further reduces to 

dx* 
The solution of this differential equation yields 
= Ax -j- B, 

♦ When the radiation toss term is included as in equation (7), 6 must be 
nuflsared as excess over ike: temperature of the surrouudinjrK. If this term is 
not included as in equation (6), S may denote the actual temperature or the 
excess over the surroundings. 









CONDUCTION OF HEAT 



CHAP 



where the constants A and B can be determined from the boundary 
mnditions. Let these conditions be (1) $ —8 at x = 0, O being 
the temperature o£ the source ; (2) 9 = 0± at x = I. Then we have 



,' 



■x, 



. - 



where is the temperature at any point x. 

If radiation losses are not negligible, the solution will be different, 
Let us assume that in this case & = e" is the solution. Hence, by 
substitution, 

n 2 — w 2 , or n = ± m. 
Therefore, the complete solution is 

= Ae™ -f Be~™ . , . . (9) 

where A 3 B are constants, 

7. Inge»-Haas56 f s Experiment*— A method of comparing the con- 
ductivity of different substances based on this solution was employed 
by Ing» •:' • early as 1789 and is generally shown as a 

' l experiment. Bars of different substance?; are coaled 
with wax and have their one end 
immersed m a hot bath of oil (Fig. 5) . 
The wax nielts to different lengths along 
different bars. Before the steady si 
is reached the U tire at any point 

depends on h, i.e., on both the thermal 
capacity and the thermal conductivity as 
in;! nation (7). This is why 

initially the temperature wave is 
i . red to travel faster along bismuth 
ii along copper, for the low thermal 
capacity of the former more than com- 
pensates for the larger conductivity of 
latter. When iJi state is attained, however, the wax is 

It on the copper over a greater length, 
Let l lf l 2 . . . . . denote the lengths along which the wax has 
melted on the differer tfc siperature of the bath, measured 

above that of the surroundings, and 8i, the' temperature of melting 
wax similarly me;. i the bars are long enough the temperature 

at their other ends is the same as that of the surroundings, i.e., 
= 0. The complete solution of this problem is represented by 
•lion (9). The boundary conditions for all the bars are (i) 0==O 
at *= co, (it) #=0o at ,v— 0, (tit) 0=0 t &tx=L 

By substitution in equation (9) condition (i) gives A = G. Con- 
';, then Lives R=0 O . Tlie solution then becomes 

Condition (Hi) then gives 

tfi-flfr-' (10) 

or ml = iog,-T— . 



tus of 

Ingen-l-j 






* 



V)il.] FORBES* METHOD 

Since log,(V*i) is tfte same £or a]1 the bars we haVC 

mik = mJ-j — rnJz =s * • • = constant, 
which from the definition of m implies that 

/ s — J a 






= s n — -\ — . . . . = constant, 



(ID 



provided the different bars have the same cross-section, perimeter 
and coefficient of emission. Thus 



K 
K m 



i _ 



(12) 



Therefore the conductivities are in the simple ratio of the squares 
of the lengths along which the wax has melted, and if the conducti- 
vity of one of the bars be known the conductivity of the others can 
be 'calculated. This is an indirect method. In order to secure the 
same coefficient of emission the bars are electroplated and polished, 

& Experiments of Despretz, Wiedemann and Franz. — Desprctz, as 
early as 1822, compared the conductivity of two substances by n 
of three temperatures at equal aces. The bars were heated at 

one end and were provided with a number of equidistant holes 
ii ( ughout their entire length. These holes contained mercury in 
which" the bulbs of mercury thermometers were immersed for record- 
ing the temperatare. The theory of the method can be worked out 
b the help of equation (9). 
Wiedemann and Fran/, following- the same principle devised 
a more accurate apparatus. The bars under test were about half a 
metre long and mi . En diameter and were electroplated One end 
of the hair was heated by steam and the remainder surrounded by a 
constant-temperature jacket. The temperatures at equidistant points 
were measured by a sliding thermo-couple which could be mani- 
pulated from outside. 




Fig. 6. — Forbes' apparatus, Statical exper' ' 

(The dpttcd curve shows fall of temperature along the bar and tangent 
to the curve jives the temperature gradient). 



186 



SUCTION OF HEAT 



CHAP 



ill. Conductivity by a Combination of the Steady 
and Variable Heat Flow 

9, Forbes' Method. — One of die earliest, methods of determining- 
the absolute conductivity of a substance is that due to Fo bes, 
Though simple in principle., it is exceedingly tedious in practice, Forbes 
used a bar of wrought iron 8 ft- long and \\ inch square section. One 
end of the bar (Fig. 6) was heated by being fixed into an iron crucible 
containing' molten lead or silver, A number of thermometers, with 
their bulbs immersed in holes drilled into the liar, were employed to 
indicate its temperature throughout its entire length. After about 
six hours the temperatures at all points become steady and are read on 
the thermometers. The temperature distribution is indicated by the 
dotted line in Fig. 6, and follows the law =&&""* *. T h 'S is called 
Lhc statical experiment since it deals with the steady state of heat 
flow. 

To obtain the heat flowing across a particular cross-section, 
Forbes determined the amount of heat lost by radiation by the 
portion of the bar lying between that cross-section and the cold end. 
These two quantities are obviously equal in the stead since 

ho heat, flows out from the cold end, this being at the room 
ture. Forbes achieved it by performing the dynamical experiment* 
so called because the temperature in this case is changing. For this. 
purpose a bar only 20 inches long but in all respects exactly similar 
to th< y'tv was used. First a high uniform teraperati 

iiinicated to this bar which is then allowed to cooi iii. exact!) 
the statical bar, and a cooling curve plotted 
for it (Fig. 7). 

; calculate the amount of heat lost by the statical 

bar from the point 
x = x- x , to the end of 
the bar (x = I) . The 
amount of heat lost 
per second by radiation 
from the surface of the 
bar from x to x + dx in 
the steady state, is 

d& 



= v 



-}-<i 




,U>:, P c w , 



Fig, 7, — Temperature curves in Forbes' bar, 



where p is the density of the material, c its specific heat and --j- 

the rate of cooling of that element. Hence the total heat lost by 
i he portion of the bar from x = « lt to x = I in a second is 

* James David Forbes (1809-1868) was Professor of Natural Philosophy in 
the University of Edinburgh from 1«J3 to i860. 



VIII.] 



CONDUCTIVITY BY PERIODIC FLOW METHOD 



187 



<-j 



Apc \w 



dx- 



This will also be equal to the heat crossing the surface at x = x lf viz., 

since the bar is long enough so that its other end is at the room tem- 
perature and hence there is no loss of heat from that end. Equating 



: i se two quantities we have 

Kldd 

P c 



:d$ 



-(f) = £*■ 



(13) 



de 



For calculating — the dynamical experiment is performed on an 

actly similar specimen with the same exposed surface. The 
observations are plotted in Fig. 7 which is self-explanatory. The 8, x 
and 0, i curves are drawn from actual observations while the values 

of— corresponding to various values of are computed from the 
0, t curve and plotted as indicated. Equation (1$) yields 

— tan <p=P, 

,■>.- 

where tan <£ and F are indicated in the figure. The area F of the 
shaded portion can be measured by means of a planimeter and hence 
K calculated. 

Tliere are several, sources of error in Forbes' method and hence 
the method fails to give accurate results. The specific heat docs not 
aain constant at different temperatures- as assumed by Forbes, Fur- 
ther the distribution of temperature inside the bar in the statical and 
dynamical experiments are different. Forbes" method has been im- 
proved by Callendar, Nicholson, Griffiths and others. 

rv. CONDUCTIVITY BY PERIODIC FLOW METHOD 

10. Angstrom's Method. — The conductivity' of a metallic bar can 
also be found by periodically heating and cooling a portion of the bar 
and observing the temperature at different times at two points along 
the bar. This method was first employed by Angstrom*. 

In his early experiments a small portion of the bar was enclosed 
in a chamber through which steam and water at any temperature could 
be alternately passed. In later experiments the end instead of the 
middle was heated. The bar was heated for 12 minutes and cooled 
also for the same time, the periodic time being 24 minutes, Texnpera- 

* Anders Jonas Angstrom (1814-1874) was Professor of Physics at Upsafa. 
He made important researches in heat, magnetism and optics. 



188 



CONDUCTION OF HEAT 



[CHAP, 



mres were observed every minute at two points along the bar by 
means o£ two thermocouples. The mathematical analysis of this case 
is somewhat complicated and will not be given here. We have to 
solve equation (6) such that the solution is a periodic function of 
time. 

11. Conductivity of the Earth's Crust— The periodic flow method 
is very suitable lor finding the conductivity of the earth's crust. The 
earth's surface is heated by day and cooled by night. This alternate 
heating and cooling effect travels into the interior of the earth in the 
form of a heat wave {diurnal wave) and gives rise to the diurnal 

Ltiona in temperature at points inside the earth's crust. Again, 
the earth receives a larger amount o£ heat in summer than in winter 
and this causes a second heat wave having the period one year (annual 
wave) which is also propagated into the interior of the earth. Assum- 
ing the waves to be simple harmonic as a first approximation (the 
annual wave.., in particular, departs considerably from this ideal state) > 

: find how they travel into the earth. The problem is thai 
bar pa iodi ally heated and cooled at one end and provided with a 

i J - — ty * 
A simple harmonic solution of (6) is 

fl = ^-** sin (orf+fi*+y) s . < - (14) 
which gives the temperature fluctuations at any point x. This is 

the equation of a 

damped ogressive 

wave and is graphically 

represented in Fig. 8. 
The wave moves for- 
ward with the velocity 
u /5 whil its a ipii- 
tude on diminish- 

ing exponentially (given 
by o ff~**) which is 
shown by the dotted 
curve. These fluctua- 
tions will be superposed on the mean temperature at any point which 
will also diminish as we go farther from the hot end. 

Tt. can be easily seen that the wave-length of the temperature 
■wave is A = — Sor/jQ. Again, by substitution from (14) in (6) we get 

—a Hf-st- • • • < 15) 

Hence a=Jv}kT. 

Now if a number of thermometers are embedded in the earth 
at different depths, the progress of the temperature wave inside 
the earth ran be investigated, and knowing the wave-length, the 
diffusivity h can be calculated from (15), In the same way we cap 
find the conductivity of any bar if we heat one end in a simple 
harmonic manner as was done by King, 



-cAX 




8L— Temperature wave at a particular instant. 



VIII.] 



CONDUCTIVITY OF THE EARTH'S CRUST 



189 



12. Applications,— The above mathematical treatment can be 
utilised for solving the deal problem, viz., the penetration of 

the dailv and annual changes of temperature within the earth's 
crust. It will be easily seen fro I '.: that the range of ajnphtucK 
R at any point is 

J2=20«0-V«/A? 
and velocity of propagation 

— ?= 2 j/f • ■ ■ < 17 > 

and wave-length A= 2yV/*T. : • - ( 18 ) 

The "lag/' i.e, } the time which any temperature applied at the 
surface talces to travel to a point x is 



v 2 (/ rt' 



,l.i. 



We shall now apply these results to the propagation of heat in 
earth's crust. Here for the daily wave 

T = 24 hours — 86400 sec. 
Taking h = 0.0049, the value for ordinary moist soil, we have 

A — 73 cm.; v = 84 X '0 * cm. /sec. ; <* = 1/11.5. 
Suppose the maximum temperature is 45°G at 2 p.m. and the 
minimum is 25°C. just after sunrise, the amplitude of temperature 
variation is, therefore, 20*C This variation will diminish as 
e ~*iiL.i at a deptli x. For x — 10 cm., *-■/»♦« = .42, for x = 30, 
tf -*ni.s_ j|-|; an d for x — 100, *-*/"••= .00018. Thus at a depth o!.' 

1 metre the temperature variations will be scarcely noticeable. 

The slow velocity. of penetration of the daily heat wave must hi 
been familiar to all observant minds in a tropical country. Here the 
roof of the house is exposed to the scorching heat of the sun. and at 

2 p.m. the temperature may be as great as G0°C. But this tempera- 
ture travels inward at the extremely slow rate of 9,1 X 10~ 4 cm./sec- 
or 6.4 cm. per hour in a mass of concrete (h — 0,0058) , Hence to 
penetrate a wall depth of 30 to 40 cm. a period of 5 or 6 hours is 
necessary. The inside of the room, provided the windows are shut, 
therefore, reaches its maximum temperature at about 7 or 8 p.m. 
when the walls become intolerably hot and begin to radiate. Most 

:. must have experienced that it is found impossible to sleep 

•rs at this time. The minimum at the top of die room is 
reached at about sunrise. So the rooms are found to be cool from 

..v£. to about 2 p.m. when the outside is hot. 

Annual Wave. — Besides the diurnal fluctuations the surface of 

arth is also subject to an annual period of 3G5£ days owing to 
the different amount of surface heating in winter and summer. This 
amount is variable in different countries, but in desert countries it 
may amount to as much as 60°C in. the sun in summer and 0°C. 
in winter. 



190 



CONDUCTION OF HEAT 



[CHAP. 



v.: ;'. 



In the case of t he ann ual wave. 

X = 73V 365 /' 1 — 73 X 19-1 CD1 - = H metres 
: 8.4 X JG~V^^ 5 c W sec - ~ 3- 9 cm - P er da y- 

1_ 
220 ' 

Thus the annual wave will penetrate a depuh o£ about 1.2 metres 
month, At a depth of x=\/2=7w. the times of the year 



fl.6 1/3(35 ~2 



» interchanged. The amplitude of temperature variation will he 
reduced by the fraction e~™>™ — ,64 at 1 metre depth and bv 
fiimt*» = ,011 at 10 metres depth. Thus the annual wave is able 
to penetrate to a depth which is V365/1 = about 19 times greater 
than the daily wave. Most of these conclusions have been verified 



wall is made up 



experimentally. 

13. Conduction through Composite Walls.— If a 

of a number of slabs of thicknesses x t , x 2 « . . and conductivities &i> 

,v joined together, the amount of heat flowing per second through 

an area A of the wall in the steady state is 



a- 



*i 



E t A 
. ,;i.re the temperatures of the intermediate surfaces and 
lr s u are the bemperaturea of Line end faces. This is because in 
ite the same amount of heat, must flow through each 
"sice 



#i 



* t -0, = (fcjJ 



0a~ fl » — Q- f~5 



A « I 



••■ 



and by addition 

b-** . - a ( jg ^ 

Therefore the heat, (lowing per second is 



a--: 



*l_+ 3 



(19) 



+ 



K\A ' K 2 A 
14. Relation between the Thermal a ad Electrical Conductivities of 



Wienemann-Franz Law.— A table of thermal conductivities 
is given on p. 200. The table shows unmistakably that all good con- 
ductors of electricity are also good conductors of heat, and even 
before any theory was proposed, Wiedemann and Franz gave the 
empirical law that the ratio of thermal and electrical conductivities 
at a particular temperature is the same for all metals. Lorenz 
extended die law and showed that this ratio is proportional to the 
absolute temperature, viz., KfaT = constant, where K is the thermal 
conductivity and a the electrical conductivity, 



vm. 



HEAT CONDUCTION IN THREE DIMENSIONS 



39r 



Drude explained this remarkable result by assuming that the 
conduction of heat and electricity in metals takes place by means of 
tree electrons. He even found theoretically the value of K/&T. 

These conclusions have been experimentally verified by Jaeger 
and Biesselhorst and by Lees. They obtained die value of K/vT 
experimentally for various metals which are given in Table L It 
will be seen that the value of K/qT remains practically contant for 
all metals. 

Table L— Values of J~X LO 8 - 
a i 





From Lees' 


experiment 


Jaeger and 
l ii: i.sclhorst 


Metal 












- 170°C. 
1.50 


- ioo°c. 


o°a 


17 D C 


10°C. 

2.19 


100°C. 


Aluminium . - 


1,81 


2.09 


2.1.3 


2.27 


Goppi 


1.85 


2.17 


2.30 


2.32 


2.29 


2.32 


Silver 


2.01 


2.29 


:'. " 


2.33 


2.36 




Zinc 


2.20 


2.39 


2.45 


2.43 


1 


2.3,3 


Cadmium 


2.39 


2.43 


2.40 


2.39 


2.43 


2.44 


Tin 


2.48 


2.51 


2.49 


2.47 


2.53 


2.49 


"J cm: 


2.55 


2.54 


2.53 


2.51 


2.46 


u\ 


Iron 


3.10 


2.98 


2.97 


2.99 


2. 70 


2.85 


Brass 


2.78 


2.54 


2/1.3 


2.45 


. , 


. , 


Manganin 


5.94 


4.16 


3,41 


3.34 


3.14 


2.97 



A closer examination of the tabic shows that the value falls 
off at low temperatures. Now at these temperatures both the 
thermal and electrical conductivities are found to increase. Hence' 
it follows that the thermal and the electrical conductivities do pot 
increase in the same ratio when the temperature falls, the electrical 
conductivity increasing much more rapidly. In fact, the latter 
appears to become extremely large at the absolute zero, if the 
metal is free from impurities. 

15. Heat Conduction in Three Dimensions. — Till now we have con- 
sidered the flow of heat in one direction only, generally along bars 
great length but small width and thickness. We shall now con- 
sider the conduction of heat in three dimensions inside an isotropic 
body. Tt can he shown by an amplification of the ideas given in 
sec. fi that for flow in three dimensions, the equation of conduction 
is given by 



ri&d , d*8 , d*9\ m 



or- 



«*-£ 



(20) 



192 



CONDUCTION OF HEAT 



CHAP. 



where h is the diffusivity. This is the Fourier equation ^ of hear 
conduction in general and we can solve it when the initial con- 
ditions are given. Equation (20) is of great importance in &tu dy- 
ing problems on heat conduction* 

V, Conductivity of Poorly Conductinc Solids 

16, In finding the conductivity of poor conductors the sub- 
stance cannot be employed in the form of long bars or rods as was 
the case with metals, for the heat loss from the sides would be 
considerable compared with the heat actually conducted away 
through the substance itself. For this reason the substance is 
generally used in the form of thin plate, sphere or cylinder. To 
this class belong- all non-metallic bodies. Cork, asbestos, clay, 
wood, bricks, etc. are among the many substances of common 
occurrence. The conductivity of these varies from 0,01 to 0.00008. 
We shall now give some of the important methods of finding the 
conductivity of these substances. 

In the methods commonly employed for finding- conductivity of 

Soorly conducting solids, energy is supplied electrically to a separate 
eating body, and the flow of heat through the experimental subs- 
tance investigated. 

17. Spherical Shell Method. — The simplest case of heat conduc- 
tion in three dimensions is that of a sphere. If the source of heat, is 

iced at the centre of the sphere the isothermal surf; ill be 

1 surfaces described about, the centre. The method has been 

employed ind others. Nfusselt's apparatus is shown in 

80 cm- in diameter and 
in;;. pper. Inside it tcentrk with it, is another hoi! 

sphere 15 cm, in diameter. The spheres can be split 

into two ted. The space between die spheres is filled with 

the material B under test such as 
asbestos, powdered cork, charcoal 
etc. An electrically heated body 
D is placed at the centre of the 
sphere inside C and electrical 
energy is supplied at a constant 
rate. Temperatures are deter- 
mined by means of thermo-couples 
:i Eh B at different distances 
from the centre and alonir one or 
more radii. Knowing- the elec- 
trical energy spent and the radii 
of the shells, the conductivity can 
be calculated. 

We shall solve the problem from elementary principles. On 
'lit. of the symmetry about the centre the isothermal sua i 
Clerical. The flux of heat across a spherical surface of radius 

r outwards in unit time is — KAfffi t-. This must equal the amount 

d r 




Fig, 9, — KTosselt's apparatus. 






VIII. J 



CYLINDRICAL SHELL METHOD 



19-5 



of electrical energy (J supplied per second to the heating body. 
Hence* 



Or 



4 77 A 



dr 



dr 



Since () is independent of r we have on integrating (21) 



8 = 



Q, 

4tt/l" 



4+* 



(21) 



(22) 



where A is a constant of integration. 

Tf the two surfaces of the shell of radii r lt r 2 acquire temperatures 

8 l3 &£ in the steady state, we have 



4irR' r t ^*' * 2 4wK 



- -4- A, 



nee 



0.= 



4fliT(^-fl Jf a 



Knowi g 0i,02 and r x , r 2 as in the above experiment the con- 
ductivity K can be calculated. Again, solving- the simultaneous 
equations for C> and A, and substituting these values of Q. and A 
in (22) , the Cempeiature distribution in the material can also be 
found, and is given by the expression 

« = r 1 r [Mrt«+ ft ~' JV 4 .* ■ (241 

18, Cylindrical Shell Method. — Let us consider the radial flow 
of heat in a cylinder which has an electrically heated wire along its 
axis. By suitably transforming equation (20) we can get the equation 
for this case. We shall, however, as before, deduce it from elemen- 
:>nsideraliom. Since the cylinder is symmetrical about its axis, 
the isothermal surfaces are cylindrical, The amount of heat {) 
Rowing per second across an isothermal surface is 



ft .- -2wrfjr=. p 



<-.-•• 



where I is the length of the cylinder. Now (J must remain constant, 
being equal to the electrical energy supplied per second. On integrat- 
between the inner and outer' radii of the cylindrical shell with 
the corresponding- temperatures B it & 2 , we have 



or 



2W" 8 x -0 t 



(26) 



The temperature & at a distance r can be shown to be given by 
9 = log (rJr ) [ ($1 l0g r *~ $ * l0g ^ _ ^i-^ lo « '") ] ■ • 



*We can also proceed as in sec. 6. 
13 






194 



CONDUCTION OF HEAT 



CHA '■ 



distance from 

en nation (2,6) 



II lias devised a method of determining conductivities based on 
this methi n er o£ the material is formed by filling 

space between two hollow concentric cylinders with that material. 
Heat is supplied by a wire carrying an electric current along the axis 
oE the cylinder, and when the' steady state is reached, temperatures 
at two points within the material are observed, as well as their 
the axis. Knowing - the electrical energy spent, 
gives the conductivity. 

simple laboratory expc 
rimeiit based on this method 
may be devised for finding fch 
conductivity of rubber and 
glass in the form of a tube. For 
rubber the arrangement shown 
iu Fig. 10 is most convenient. 
Steam from the boiler A tra- 
verses through a rubber I 
H t a length J! of which is 
mersed in a weighed amount of 
water contained in the calori- 
meter C. The radial flow of 
heat Q_ from the tube to the 
water in C is given by 




Fig. 10,- -Apparatus for finding 
conductivity of rubl 



log. 



(28) 



re $i n the temperature of steam, and {.)-, the mean of 

the initial and final tern] ■ of the calorimeter. This heat can 

in temperature of the calorimeter and its 
id hence equating these two quantities we get 
it*. 
For ;emenl shown in Fig. 11 is convenient. 

is tube C has a spiral wire along its axis and is surrounded by i 
mi jacket, while a steady current of water flows through the tube.\ 
The method of calculation is similar to the preceding one. 




Walt,- 
Sieam 
W ate * 

Fig. lb— Apparatus for finding conductivity a£ glass. 

19. Lees* Disc Method* — Another convenient method for finding 

conductivity of a very bad conductor has been given by Lees 

♦For experimental del- Worsnop and Flint, /Vc ■'■. asks. 



DHL] 



LEES' DISC METHOD 



J 95 



Steam 



in which the substance is used in the form of a thin disc. The 
apparatus (Tig. 12), due to Lees and Charlton, consists of a cylindrical 
steam chest A, the bottom of 
which is a thick brass block B in 
which a hole is bored for inserting 
a thermometer T lf The substance 
S (shown shaded), En the form of 
» a circular disc, is sandwiched 
between the block B and a second 
cylindrical brass block C, the latter 
carrying a second thermometer 
I s . The I^ck C is suspended 
m a retort stand by threads so 
that its top face is horizontal. 

The radius and the thickness d 

of the specimen are first measured. 




Fig. 12— Lees' disc method 
f f bad conductor. 



The apparatus is then set up as shown in Fig. 12 and steam is 
d through A for a pretty lone time until the readings in the 
thermometers T 3 and T 2 are steady. These temperatures can be 




received by it from conduction through the specimen must be equal 
to the rate at which heat is lost by it by radiation from the sides 
and the bottom surface of C. The former is given bv 

G-JM&^&, . . . (29) 

where A' is the required conductivity and A the area of the disc. The 
latter is determined in the following iva; ; 

The block C h alone heated separately by a bunscn burner until 
its temperature rises to about !0°C above 2 . It is then alone sus- 
pended with the specimen placed on the top and allowed to rool. 
Its temperature is noted at regular intervals of time until it cools to 
about 10°C below g 2 and a cooling curve drawn, From the tangent 

to the curve at 8 2 , the rate, of cooling - f at 0., is determined. The 
rare of loss of heat by cooling is 



a— 'GL 



. (30) 

where m and s denote respectively die mass and the specific heat of 
the block a Combining (29) and (30) , K can be found. 

In die above we have assumed that no heat is lost from the 
curved surface of the specimen, and consequent! v the specimen must 

ery thin. 

The disc method can also be used with electric heating when a 
modified arrangement is necessary. A thin plate of the experimei] 






J 96 



CONDUCTION OF HEAT 






substance is placed between two copper plates, A coil o£ insulated 
wire is place above die tipper copper plate and held down by another 
copper plate as explained later in § 22. Electrical energy is supplied 
to the coil and steady state obtained. The amount of heat passing 
through the experimental disc can be obtained from the electrical 
energy supplied and the loss of energy due to surface emission. 
Knowing cue temperature of the copper plates the conductivity 
be calculated. Alternatively, the two-plate system can be employed 
by placing above the heater coil another sandwiched experimental 
plate, exactly similar to the system below the heater coif. In this 
arrangement it is not necessary to know the radiation loss as all the 
electrical energy is transmitted through the two specimens. 

CONDUCTIVITY OF LIQUIDS 

20. The determination of the conductivity of a liquid is 
complicated by the presence of convection currents. If we heat 
a column of liquid at the bottom, the liquid at the top receives 
heat both hy conduction and convection. The laws governing- convec- 
tion currents are complicated] hence it is preferable to eliminate them. 
This is accomplished either by taking a column of liquid and heating 
it at the top or by taking a thin film of liquid. 

21. Column Method. — The column method was employed by 
Dcspretz long ago. The liquid at the top was kept heated and tem- 
peratures axis ot the column were observed by mercury 

He Found that Fourier's equation already derived for 
a bar hold? true in this case riiso, and hence the conductivities of two 
Liquids caj with the help of equation (11). Weber sur- 

fed the column with a guard-ring and used as the source of heat 
an electrically heated oil bath. The bottom of the column was cooled 
copper plate standing in ice. The heat conducted away was 
found from the amount of ice melted. Knowing the temperature 
bution along the column the absolute conductivity can be calcu- 
lated in the same manner as for mercury in Berget's experiment. 

22. Film Method,—- This was employed by Lees>, Mihier and 

Chat Lock and by Jakob, We 
shall describe the apparatus 
iiied by Lees which is simply 
a modification of his disc 
method for finding the con- 
ductivity of poor conductors. 
The liquid L under test is 
enclosed in an ebonite ring E 
(Fig. 13) and placed between 
copper blocks, C^ C.,. To de- 
termine the quantity of heat 
flowing through the liquid a 
glass disc G is inserted, above 
which another copper plate 
Above C 3 and insulated from 




Fig, 13. — Lees 1 apparatus for finding 
the rouductivity of liquids. 

C 3 is cemented with a layer of shellac. 






CONDUCTIVITY OF GASES 



197 



it by mica is placed a flat spiral coil of heating wire W which is held 
down by another copper plate C. The whole pile is varnished and 
enclosed in an air-bath. Temperatures are recorded by thermocouples 
soldered to the faces of the copper plate. The calculations can be 
easily made. 

Let Si, S s , Sg, S a denote the emitting surfaces of CY, C 2! C s 
and G respectively, h their emissivity and T-,., T„, T & the temperatures 
i I' the copper discs. The heat passing through the middle section 
of G is by definition equal toai£T G (T a - F a )/d . Of this the amount 

9 ' —^o — 2 i s I° 5t fry tne lower half of G by radiation from the sides 

and similarly S 3 h T,-. is lost by C s . Hence the heat transmitted 
through the liquid and the ebonite is equal to 



AK t 



T 9 -r t 



— SqH ' -'— -, — - — $tkT s 



and thus must evidently be given by 



: B(T,-T i ), 



some constant giving the transmission of energy through 
the ebonite E for unit difference of temperature. Equating these two 
expressions we get K& in terms of K G provided B is known. B can 
be determine! ::rking with air whose conductivity is known. 

23. Hot-wire Method. — Goldschmidt employed the "hot-wire" 
method of Andrews and Schleiermaeher, the theory of which is dis- 
cussed in sec. IS. The liquid was contained in a silver capillary Lube 
2 mm. in diameter heated by a wire running along its axis, 

CONDUCTIVITY OF GASES 

24, The determination of the conductivity of gases is difficult for 
the phenomenon is always accompanied by radiation, and sometimes 
by convection currents also. Kundt and Warburg showed that the 
rate of cooling of a thermometer immersed in air remained constant 
for pressures lying between 150 mm. and 1 mm. Hence the effect 
of convection currents is negligible in this region and heat is lost only 
by conduction and radiation. To determine the radiation loss, the 
air was exhausted as completely as possible, and then the rate of cool- 
ing was found to be independent of the size of the enclosure, showing 
diaf the effect of conduction was negligible and heat was lost only 
by radiation. Subtracting this radiation" loss, we get the heat lost hy 
conduction alone. For finding the conductivity of a gas at pressures 
higher than 150 mm. the gas may be exhausted to this pressure (be- 
tween 150 mm. and. 1 mm.) and its conductivity determined. Now 
since the conductivity of a gas is independent of the pressure (see 
Chapter III, Sec. 24) this will give the required conductivity. Another 



198 



COXDUCriON OF HKAT 



[CHAP. 




Fiff. 14. — A '_:■•.: 'd'atus to sllf>w the 
better conductivity of hydrogen- 



procedure consists in taking a thin, film of gas and healing it at the 
top when convection will be absent 

A very simple experiment 
described by Andrews and Grove 
allows qualitatively that, hydro- 
gen is a far better conductor of 
iat than any other gas, A fine 
latinum wire was supported in- 
side a glass tube -(Fig. 14) which 
could be filled with any gas. 
. wire could be heated by an 
electric current and made to 
glow. Two such tubes were 
; ranged side by side, one filled 
with air and the other with 
hydrogen. The same electric: 
current was allowed to I 
through both the wires. The 
wire in the air-tube can be 
glowing while the wire in the hydrogen tube does not glow at all. The 
tss tube containing hydrogen also becomes hot. Heat is very quick- 
ly conducted awa '■■ and hence the wire is not raised to 
the temperature of incandescence-. Replacing hydrogen with air res- 
totes the incandescence. 

[<e important methods of finding the a i xmductivity of a 

■ 

Method, (2) Film Method. 

25, Hot-wire Method. — This method, first given by Andn 

emi. acher lor determining the absolute conducti- 

vity electrically heated vertical wire is surrounded by 

a coaxial c filled with the experimental gas. The temp. 

e of the wn:e is known from its resistance, and the amount of heat 
flowing across the curved surface of the cylinder is found from the 
rate of energy supplied to the wire. The conductivity can be calculat- 
ed from equation (2ft) where r t , r s now denote the radii of the wire 
and the tube respectively. In this arrangement convection is very 
much minimised sine:: there is no temperature gradient in the vertical 
irei ion, Radiation losses and whatever convection losses remain, 
eliminated as pointed out. in sec. 24. Correction must, however, 
be applied for loss of heat by thermal conduction along the leads 
supplying the electrical current. The method has recently yielded 

■• accurate and reliable values for a large number of gases. 

26, Film Method, — The film method, originally .used by Todd, 
has been employed by Hereus and Laby. The principle underlying it 
is th^ same as in Lees' experiments (sec. 22) . The thin film of gas 
under test was enclosed between two copper plates B and C (Fig, 15), 



viu.'l 



CONDUCnvrTY OF GASES 



199 



the latter of which was cooled by a current of water. The upper plate 
B was made up of two sheets of copper clamped together with a heat- 
nl between them. To prevent loss of heat by radiation from 
the upper surface of B, there was another plate A above it at the same 
temperature and a guard-ring D surrounding it. On account of the 
latter the flow of heat from 1> to C was linear. The plates A, B, D 
separate heating coils and thermocouples and were kept at 
lame temperature. The thermocouples- were formed by attaching 
untan wires to the copper plates, each plate having a copper- 
lead also attached to it. The plates were accurately ground and silver 
plated. The whole apparatus was made air-tight by a ring of rubber 
clamped to A and C by steel bands. The temperature of A was gener- 



Ebanitc 




BcaiS 




iV iier 






. IS.— Apparatus of Hereas and Laby. 

ally kept a little above that of B in order to eliminate :; ability 

onvection. This, however, necessitated a small correction. The 
radiation correction was determined by a separate experiment on a 
silvered Dewar flask and was only about 5 per cent. Convection 
effects arc: entirely absent since tlie gas is heated at the top, The 
energy spent in B was known electrically and subtracting from if the 
lost by radiation, the heat transmitted to C by conduction through 
the air film wis found. Knowing the temperature of C the conduc- 
tivity of the gas can be calculated from equation (1) . 

27. Results, — The thermal conductivity of a number of sub- 
stances is given in Table 2, p. 200. The value of K is given in 
calorie cm.- 1 sec.-? C- X * 

It will be seen from the table that silver is the best conductor of 
heat (K — 1) and copper comes next. The conductivity is less for 
liquids and least for gases. The conductivity of gases is extremely 

(of the order of 1CH 3 ) . 

We have: already considered on p. 85, Chap. Ill, the relation 
between thermal conductivity and viscosity of gases, as well as the 
variation of the thermal conductivity of gases with pressure. 



200 -lUCTlON OF HI ■ ■■. : [CHAP, 

Table* 2. — Thermal conducti of different tances. 



Substance 




Metals (0°C.) 



Aluminium , , j 


. 


Cadmium . - ! 


0.23 


Copper 


0.93 


Iron (pure) 


0.16 


Lead 


0.085 


Mercury 


0,02$ 


Nickel 


0.1-1 


Silver 


1.0 


Platinum 


0,17 


Tin 


0.155 


Zinc 


i. :.r:.< 



Asbestos paper 
Cardboard 
Cork (p=.16) 

Paper 
Ebonite 
Mica 
Paraffin wax 

Pine wood 
Rubber 



Alloys (0°C.) 



Brass 

an 
Mauganin 



0.05 
0.06 



Poo\ 



1 lint glass 



2.5x10- 

2 



Liquids 



Water (24°) 
Alcohol (25 D ) 
Glycerine (25*) 



0.6k 10" 

0.5 

0.11 

0.3 

0.42 

1.8 

0.6 

0.4 

0.45 



14.3XJ0 4 
4.3 



jes (0°C.) 



Helium 

Argon 

Air 

<:Ll 

Carbon dioxide 



10 ■* 

3.89 
5.40 

5.63 
8.07 



28. Freezing of a pond.— An interesting example of conduction 
across a slab of varying thickness is provided by the phenomenon of 
freezing- of water in ponds and lakes during winter. When ice I 
to form on a pond, the bulk of the water in the pond is at aboui 
while the top layers are at 0°G, and the cold air above abstracts the 
latent heat from a narrow surface layer at 0°C. The subsequent 
growth of rile ice layer requires that the necessary abstraction of 
latent heat take* place by conduction through the layer of ice already 
there. Assume that, the thickness of the ice layer already formed is 
z, and die temperature of the water below this layer is 0°C. Then 
for a further freezing of the layer of area A and thickness dz in the 
time dt 3 the quantity of beat 

Q = Apdz.L 

-Taken partly from Landolt and Bernstein, PhysikalUh-Chemischen Tabelten 
and partly from Kaye and Laby, Tables of Physical and Chemical Crttwttents, 



CONVECTION Ol 7 HEAT 



201 



vii i.] 

uu-.l travel upwards by conduction through ice of thickness t, wner 

P = density of ic& and L — the latent heat of fusion- Hence 

where K = thermal conductivity of ice, and -£ is the temp era Cure of 
the aiT above the pond. Equating we get 

pL 
which on integration yields 

P L 
Lhe constant of integration being zero as z = at * = 0. Thus the time 
10 obtain a given thickness is proportional to the square ot ttie 
thickness. 

Exercise.— The thickness of ice on a lake is 5 cm. and the tem- 
perature of the air is -2G°C. Find how long it will take for the 
thickness of the ice to be doubled. (For ice thermal conductivity is 
0-005 cal cm- 1 sec.- 1 C 'C~\ density = 0,92 gm./cx,, latent heat -= 80 
cal./i i 



[\Ve have 



*K-SH 

IpS-a'-* pL 
t= = 2 KB 

10'— 5* 0-92x80 
~2 0405x20 

= 27,600 sec = 7 A 40 rtt .] 



CONVECTION OF HEAT 

29. Natural aad Forced Convection*— We have already stated in 
§ 20 and § 24 that, there will be transfer of heat by convection in fluids 
unless proper precautions are taken. Convection is the transference 
of heat by heated matter which moves carrying its heat with it. Thus 
it can take place only in fluids. Free or natural convection always 
takes place vertically and is caused by gravity as a consequence of 
the change in density resulting from the rise in temperature and con- 
sequent expansion. ' In forced convection a steady stream of fluid 
is forced past the hot body by external means. Still-air cooling is 
ural convection ; ventilated cooling in a draught is forced convec- 
tion. 

The theoretical treatment of convection is rather complicated 
though the problem of forced convection is a little simpler. Never- 
theless convection is of great practical importance. The land and 



202 



CONDUCTION OF HEAT 



[CHAP. 



sea breezes, the trade winds, the tall of temperature with height in 
the atmosphere (discussed fully in Chap, XIT) are all examples of 
convection on a huge scale in nature. The ventilation of rooms and 
the central heating of buildings in winter arc some examples of forced 
convection in e very-day life. 

30, Natural Convection, — 'When a heated body is cooled in air. 
all die three methods of heat transference are acting sirimltaneoush. 
But the air is a very poor conductor of heat, and radiation is import- 
ant only for large differences of temperature ; thus the chief means of 
heat loss is convection. The mechanism of this heat loss is easy to 
understand but the derivation of a theoretical formula is extremely 
difficult. The problem is therefore best studied experimentally or 
by a recourse to the method of dimensions. 

In [lie film theory of cooling by convection the mechanism of 
heat loss is somewhat as follows :— It is believed that the whole sur- 
face of the cooling body is covered by a thin layer of stagnant fluid 
adhering to the surface. In natural' convection this film is perma- 
nently present while in forced convection this is being continuously 
wiped off and renewed, Thus in natural convection the heat has to 
flow across this film of air and the amount lost will depend upon the 
thermal conductivity of the air and the temperature difference between 
the body and the air. The heat transmitted will raise the tempera- 
ture of the air > convection which will he opposed by via • 
forces. Thus the heal loss will depend upon the specific heat, expan- 
nt and viscosity of the air. 

experiments on natural convection was 

edit. They observed the rate of cooling oi 

ii :i large coj iper globe surrounded by a hath at a fixed 

riments were first, done with the globe eva- 

when the heat loss w is due to radiation alone and the rate ■■ 

fall of temperatui Bound to be 



-5-*c-^>. 



(31) 



where ' a constant depending upon the nature of the surface and 
Q, 8 the temperature of the thermometer and the enclosure res- 
pectively. Next the globe was filled with different gases at different 
pressures, and rate of cooling observed. Subtracting from this the 
radiation loss, the loss due to convection alone was computed and 
found to he 



-^^nipHd-6^*. 



m 



where p is the pressure of the gas and nij c are constants depending 
upon the gas, Thus the rate of loss of heat due to natural convection 
can he written in the form 



- f= k(d-o )^ 






VL1J. 



CONVECTION OF HEAT 



203 



where k is a constant depending upon die gas and its pressure, and 
5/4 ha^been written instead of 1.233. This Js called the five-fourths 
power law for natural convection which can be also deduced theore- 
tically. 

For forced convection experiments show that the rate of loss 
of heat is proportional to die temperature excess (<?-0 o ) . It will be 
thus seen that Newton's law of cooling holds for forced convection 
even for large temperature differences but is not true for natural 
convection. As shown in Chap. XI it holds for radiation provided 
the temperature difference is small. 

Books Recommended 

1. Glazebrook, A Dictionary of Applied Physics, Vol. T. 

2. Ingersoll and Zobel, Mathematical Theory of Heat Conduc- 
tion (Ginn, 1913). 

3. Carslaw and Jaeger, Conduction of Heat in Solids (10 : 
Clarendon Press, Oxford. 



CHAPTER IX 
HEAT ENGINES 

L Introduction to Thermodynamics*— Thermodvn amirs is liter- 
al I j the science that discusses die relation o£ heat to mechanical 
energy. Rut m a broad sense, in comprises die relation of heat to 
other forms of energy also, such as electrical and chemical energy, 

it energy, etc. The principles of Thermodynamics arc very gem 
m their scope, and have been applied widely' to problems in Physics, 
Chemistry and other sciences. The theory of heat engine* r| ; Q 
an integral part of the subject, and as the 'early developments were 
largely in connection with the problem of conversion of heat enemy 

S S eC ^ 1C W ?™' we sha . 1] ^gin the study by devoting a chapter 
to the Theory of Heat Engines, 

HEAT ENGINES* 

2. The progress of civilisation has been intimately bound up 
with mans capacity for the development and control of power 
History tells us that whenever man has been able to make a great 
discovery leading to a substantial increase in his power, a fresh epoch 
m civilisation began, * 

he present ag sometimes been styled as the 'Steam Aee' 

1 influence exerted by the invention of the 
^ of human proves, Tn this chapter. 
u make a brief survey of this "great event." 

present time we know that Heat is a kind of motion 

r motion disappears it reappear, as heat, and experiments 

hat 1 calorie of heat is equivalent to 4.18 x 10* ergs of work. 

e question naturally arises : "Can we not reverse the process? Can 

we not by some contrivance, convert heat which is in so much excess 

;»t m, to useful work?" This is in fact the function of heatengines 

to wo r r arE g ° ing t0 They "* contrivailces to convert heat 

"N»m£ J? wJHSft to «*» to » u jw« Profoundly ignorant of the 

irfrhlffnl 4-t he T thC Pr0 ? em did not F* 6 * *** to them 

! r t,^ Z hey \ bwever ' ? bseiTed that B«wDy when bodies 

verTbodv C ?' P P ° Wer " D ° ay take three exa Wfe familiar 

t™ ,!; When WaL T i 5 ^ iled in a dosed ketdc > rhe lid is blown off 
place to acknowledge our grateful thanks to the author. 



TX.j 



EARLY HISTORY OF THE STEAM ENGINE 



205 



Z When gunpowder or any explosive is exploded a sudden im- 
pulse is created which may be utilised for throwing stones, cannon 
balls and for breaking rock. 

3. High velocity wind can be made to do work, e.g., from early 
times sails have been used for the propulsion of ships, lor driving 
mills (wind-mills) . We know that such winds arc due to intensive 
heating of parts of the earth's surface by the sun. 

The three illustrations chosen above have served as the starting 
point for three different classes of engines which convert heat to work, 
viz : (I) the steam engine widely used for locomotion and in in- 
dustry, (2) the internal combustion engines used in motor cars, aero- 
planes, and for numerous other purposes, (8) the windmills and steam 
and gas turbines.* 

Many of the principles utilised in these engines art mute common 
to all classes, and we shall begin by describing the evolution of the 
steam, engine. Though the mechanical details" are outside the scope 
of this book, an elementary discussion is included for the sake of 
completeness and continuity of treatment. 

3- Early History of the Steam Engine. — The earliest record of 
human attempt to make a heat engine is found in the writings 

Hero of Alexandria, a member of the famous Alexandrian school of 
philosophers (BOO B.C.— 100 A.D.) which included such famous men 
of science of antiquity as Ptolemy (astronomer) , Euclid (geometer) 
and Eratosthenes (geographer) . Hero describes a scientific toy in 
which air was heated in a closed box and allowed to expand through 
a pipe into a vessel below containing water. The water was thus 
forced up through another pipe into a vertical column producing an 
artificial fountain. There was, however, no suggestion to employ it 
on a large scale. In 1606, about two millenia after Hero, Marquess 
Delia Porta, founder of the Neapolitan Academy and one of the 
pioneers of scientific research in Europe, employed steam in place 
of air in Hero's experiment in order to produce a fountain. He also 
suggested that in order to fill up the vessel with water, it may be 
connected by a pipe to a water reservoir below. If the vessel filled 
uo with steam be now cooled with water from the outside, steam inside 
will condense, a vacuum will be produced, and water will be forced 
up from the reservoir, replenishing the vessel again. This principle 
was utilised by Thomas Savery in 1698 to construct a water-pump in g 
machine which is described below. He was the first man to produce 
a commercially successful steam engine which was extensively used 
for pumping water out of mines, and supplying water from wells. 

The principle utilised in Lhis engine is illustrated in Fig, 1 (p. 
206) . V is a steam boiler. A, B and" C are valves, The operation 
takes place in two stages * — 

♦Recently during the second world war a new type of engine based on the 
rocket principle was developed in Germany and Italy. In these a high velocity 
jet of air escapes at the rear of the machine which on account of the reaction 
thus produced moves forward with tremendous velocity, 



T 



206 



HEAT ENGINiS 



j ( ■. i \r 



ix.l 



J AM liS W> 







Fig. 1. -Principle cf Savory's 



(a) B is kept, closed and A. C are open, Steam passes from V 

to P and forces tlie water Eg 

(b) A and G are closed, and 
B opened. Cold water is sprink- 
led on P, This condenses steam 
in P, a vacuum is created and 
water is sucked up from the pit 
E to P, After this the operation 
(a) may be again performed and 
a fresh cycle begun, 

Savery*s engine could not 

suck water through, more than 34 

Irt'i,, but it could force up the 

water to any height. In fact, 

be sometimes forced up water to 

a height of 300 feet. This means. 

that he used high pressure steam 

up to 10 atmospheres. This was 

a risky procedure though Papin had shown about 1680, how the risk 

in using high pressure steam could be minimised by the introduction 

of the safety valve. 

Papin, a French settler in England, had discovered a method of 

softening bones by boiling them in a closed vessel under pressure. 

as we know, raises the boiling point of water to about 150°C 

and makes the watei a very powerful solvent, Papin invented the 

boiler to re vent his vessel from 
being blown up by high pressure steam. This is shown in Fig. 2. 

valve consisted of a rod LM 

ted at I. and carrying a weight N at the 

nd. Tt pressed down the valve P which 

exactly fitted the top of the tube HH leading 

from the inside of the boiler. Whenever the 

mii pressure exceeded a certain limit, it 

forced up the valve P and the excess steam 

would rush out. By adjusting the weight of 

or its distance from L, the maximum steam 

isure could he regulated at will. 

4. Newcomen'3 Atmospheric Engine* — The 

next forward step was the invention of Newco- 
men's Atmospheric Engine which was designed to 
pump out water from mines and wells, and was 
in practical use for more than fifty years. This 

ie is interesting from the historical point of view since it directly 
led to the grear inventions of James Watt, and it employed for the 
first time, the cylinder and the piston, which has been a feature ol 
steam engines ever since. Fig. 3 illustrates the Newcomen Engine. 




Fig. 2.- iPap&i'g 
Safety Valve. 



A is the cylinder, T is the piston suspended by a chain from the 
lever pivoted to masonry works. The other arm of the lever carries 
the piston rod W of the water pump which goes into die well. There 
is a counter-weight M to balance the weight of the piston T. The 
problem is to move the piston T up and down. 

This was achieved as follows : — Starting with the piston T at 
the bottom of A, steam is 
introduced from the boiler 
B which forces the piston up 
till it reaches die top. The 
steam is shut off by the tap 
1), and cold water sprayed 
through F which condenses 
the steam in the cylinder. 
Vacuum is produced inside 
the cylinder and consequ- 
ently the atmospheric pres- 
ide forces down the piston, 
I J is again opened and a 
fresh cycle begins. The 
water in the cylinder A 
drains out through a side 
pipe. 

For closing and open ing 

the valves automatically, a 

allel motion guide was 

provided which carried 




Fig-. 3. — Nciwcomeir'.-, ALii:-:.ii.u-. lie 
Engine. 






mechanism lor automatically operating the valves. The story goes 
that the invention was due to a lazy boy who was employed to close 
and open the valve by hand, but who tied a parallel rod to the swing- 
arm of the lever, and connected it bv means of cords to the valves, 
and leaving this rod to do his work enjoyed himself all the while in 
playing. Whatever may be the origin, the parallel guide has been a 
permanent feature of steam engines ever since. 

In the Newcomen Engine, the useful work is done by the atmos- 
pheric pressure while steam is only employed to produce vacuum, 
hence the name atmospheric engine. It is easily seen that it is very 
wasteful of fuel. 

5. James Watt. — James Watt is commonly credited with the- 
discovery of the steam engine. The circumstances which directed 
his attention to steam engine are pretty well known. He was an 
ingenious scientific instrument maker at Glasgow, and in 17G.-5 he 
asked by the professor of Physics at the Glasgow University to 
: I .:• a mien Engine belonging to them which had never worked 

well. Willie i : : i d in the' repair of this machine, the idea occurred 
to him that the Newcomen Engine was awfully wasteful of fuel, and 
being of an inventive temperament, he began to ponder and experi- 
ment on the production of a better type of machine, He was thus 












f ENGINES 



CHAP- 



led to a series of investigations and contrivances which gave the steam 
engine its present form and rendered it a mighty factor in the onward 
march of industry and civilisation. We are describing some o! 
inventions below. 

6. Use of a Separate Condenser,— Watt observed that a large 
part of the expansive power of steam is lost on account of the fact 
that the cylinder is alternately heated and cooled. The expansive 
power of steam depends upon its temperature. Now when the steam 
enters the cylinder, which has been previously cooled to create a 
vacuum, sonic heat is taken up by the cylinder in becoming heated 
and is not converted into useful work. The temperature ot steam 
falls and its expansive power is diminished. Another disadvantage in 

using the cylinder as condenser is that cold 
water entering the cylinder becomes heated 
and exerts appreciable vapour pressure. 
thus preventing the formation of a goad 
vacuum. The "problem was to cbndensi 
the steam without, cooling the cylinder. 
Watt achieved this by the use of a separate 
condenser. 

The principle of the separate conden- 
ser is illustrated in Fig. 4. A A is the 
cylinder in which the piston P moves -to 
and fro. The piston is provided with a 
; i n he PQ carrying a. valve Q at the 
nser - end such that Q allow* steam to go out 

,ni is closed by the atmospheric pressure when there is vacuum in- 
Eng with the piston P at the bottom of the cylinder, R r 
j are and the part of the cylinder above P is filled with 

nit the air and residual steam through Q. Then S 
id T opened. The steam ■ Lwn into the condenser C 
which had been slv evacuated by the pump B f and is there 

ilensed by the cold water surrounding the condenser. Come- 
ly a vacuum is produced above P and steam from below pushes 
die piston P upwards, doing work on the weight. W- Then T is 
closed. S opened and P is drawn down to the bottom by W and the 
cycle begins afresh. The pump D serves to remove the air and water 
produced from steam in G. 

To keep the cylinder hot, Watt further surrounded the cylinders 
by a steam box and wood. Now-a-days the cylinders are jacketted 
tth asbestos or some badly conducting substance, and then covered 
with thin metal sheets. 

7. The Double-acting Engine.— In the Newcomen engine we 
have seen that the atmosphere pushes down the piston. Shortly 
afterwards Watt employed steam instead of the atmosphere to pull 
the piston down. The raising of the piston in the subsequent stroke 
was brought about by a counter-weight attached to the other arm of 




To face p. 208 




Watt 



(p, 207) 



James Watt, bora in Scotland in 1736, died in 1819. His important 

work is the masterly perfection of the steam engine which 

Increased the powers of man ten times and ushered 

the 'Industrial Revolution." 



IX.] 



t.i-lLIo.VlIOX OF EXPANSIVE POWER OF STEAM 



m 







Cabnot (p. 213) 

Nicolas Leonard Sadi Carnot, bom on June 1, 1796 in Paris, 

died of cholera on August 24, 1832, He introduced 

the conception of cycle, of operations for 

heat engines and proved that 

the efficiency of a reversible 

engine n maximum 



the beam. For these operations lo be possible the upper end ol I 
cylinder must be closed. Watt achieved tiw ant o a steam- 

efat stuffing box which is full of oily tow. This is kept Ughtly press- 
ed 3 against trie piston so that the piston can move through the cover 
without loss am. This was the so-called smgk-actmg 

Wat i Watt, however, soon realised that in this engine no 
done'hv steam dining that stroke in which the piston was raised up 
by the "action of the counter-weight. He saw that the power could 
be a mately doubled if during this useless stroke* steam is aa- 

mitted to the lower side and the upper side is connected to weefflj- 

i ;er. This is achieved in the double-acting engine, invented by Watt, 

with the aid of a number of valves, A modern double-acting cylinder 

. is shown in Fig, 5. The cylinder has ports or holes 

\, B, near its each end and between these lies another port E leading 

the exhaust or condenser. To the cylinder is fastened the 
steam chest C containing the It-slide valve S. Steam from the boiler 
enters the steam chest at the top. In the position (a) steam enters 
the cylinder through the port B and pushes the piston to the left, 
thereby driving the steam in front through A to the exhaust *,. AS 
the piston moves to the left the slide valve moves to the right and 




. 5.— Double-acting cylinder with slide valve. 

closes both the ports A and B For a time, and later when the i 

v.-* the extreme left position, B is closed and A opened. _ Steam 
then enters through A forcing the piston backward and > ng the 



The double- 
kinds of steam 



then enters through A forcing the p 

steam in front to the exhaust. This is shown at (b) , 

acting engine is now universally employed in all 

engines. ,-,.,-, r 

The timely action of the slide valve is adjusted by means of an 
entric wheel attached to the moving shaft (see Fig. 7). 

powerful engines as in locomotives the slide valve is often repla< 

by a piston valve which is very similar. 

8. Utilisation of the Expansive Power of Steam.— Watt's another 
at inventi .the so-called expansive working of steam. 

saw that if steam is allowed to enter the cylinder all the time 

n is movi • tfards, the steam pressure in the cylinder wih 

„e the sac i the boiler and though we get a powerful stroke, 

the expansive power of steam is not utilised If, however, the steam 

14 






210 



HEAT ENGINES 



CHAP. 



is cut off when the piston has moved some distance, the piston would 
complete its journey bv the expansive power of steam, whose pressure 
will in consequence be reduced to almost that of the condenser. Thus 
more work is' obtained from the same amount of steam by allowing 
the steam to expand adiabatically and hence the running of the 
machine becomes considerably economical. It is thus of great 
advantage to use high pressure steam. 

It was mentioned in the last section that the slide valve closes 
bo lit the ports A and B when the piston has moved some distance. 
From this instant to the end of the stroke the steam is allowed to 
expand adiabatically. 

9. The Governor and the Throttle Valve.— Another simple but 
very useful invention of Watt was that of the governor. This is a 
piece of self-acting machinery which controls the supply of steam 
from the boiler into . . cylinder, and ensures smooth running of the 

engine at a constant spe 

Watt's governor is shown in 
Fig. 6. S is a vertical spindle 
which is made to revolve by 
means of gearing from the en- 
gine shafts. It* speed, there- 
fore, rises or falls with the 
engine speed. It carries a pair 
of heavy balls which are fasten- 
ed to S by rods pivoted at P. 
The balls rise on account of 
centrifugal force as the spindle 
rotates, and as they do so they 
pall down a collar C which slides 
smoothly in the spindle S. The 
ne end of a lever L, pivoted at Q. The other end 
p in the steam ipe called the throttle valve. As 
ulled down, thi ■ Live tends to close the tap, the steam 

a] md the engine speed falls. If the engine speed is too 

diminished, the balls fall down, C is pushed up, and the throttle 
valv. tting more steam, and the speed goes up. Thus the 

governor automatically regulates the speed at which the engine runs. 
Improved forms of governors are now employed in engines. 

10. The Crank and the Flywheel— Watt was the first to convert 
the to-and-fro motion of the piston into circular motion by means o£ 
the connecting rod and the crank. Thus the steam engine can he 
made to turn wheels in mills, work lathes and drive all kinds of 
machinery in which a rotary motion is needed. 

The connecting-rod R and the crank C are shown in Fig. 7 
at (a). The crank is a short arm between the connecting-rod 
ant! the shaft S. The connecting-rod is attached to the piston rod 

consequently takes up the to and-fro motion of the latter. As 

ing-rod forward it pushes the crank and thereby 

rotates the shaft S. In the return stroke the circular motion is. 




(Ill-: t 



IX.] 



MODERN STEAM ENGINES 



211 




I'iij. 7. — Crank, Eccentric and* Flywheel. 



completed. There are., however, two points in each revolution when 
the connecting-rod and the crank are in the same line and the pis- 
ton exerts no turning moment. 
These are called the 'dead 
centres/ At two points when 
the crank is at right angles to 
the connecti ng-rod the torque 
is maximum. To prevent the 
large variation in the magni- 
of the torque producing 
variations in the speed of the 
shaft during a single revolu- 
tion a big flywheel F, shown 
at (c) s is attached to the shaft. 
The flywheel on account, of its 
large moment of inertia carries 
the crank shaft across the 
dead centres ', in fact, it ab- 
sorbs the excess of energ) sup- 
plied during a part of the half- 
revolution and yields had I 
same in the remaining part of 
the half-revolution when less 

energy is supplied. Thus the flywheel acts as a reservoir of energy 
which checks variations during a single stroke, while the governor 
prevents variations from stroke to stroke. 

Another mechanism to convert the to-and-fro motion into cir- 
cular motion or vice versa is the eccentric, shown at (b) , Fig. 7. 
It consists of a disc mounted off its centre on the shaft S and sur- 
rounded by a smoothly fitting collar to which the rod is attached. 
The behaviour is as if there was a crank of length SC. . Such an 
eccentric is mounted on the shaft carrying the flywheel (shown at c) 
and works the slide valve. The effects can be properly timed by 
suitably mounting the eccentric on the shaft. 

The essential parts of a simple engine are shown in Fig. 8. 
They will be easily followed from the figure. 

11. Modern Steam Engines. — Since the time of Watt many 
important innovations have been introduced into the steam engine 
though the main features remain the same. The innovations were 
needed in order to suit the circumstances of ever- widening applica- 
tion of steam engines to various purposes. Watt always used steam 
engines with low steam pressure., and of a static type. He was 
evidently afraid of explosions. But engines using low pressure steam 
are comparatively inefficient, as we shall see presently, and in modern 
times high pressure engines have almost replaced the old Watt 
engines necessitating the construction of special type of boilers. 

Condensers in modern engines consist of a number of tubes 
containing cold water kept in circulation by means of a pump, and 



JT 



212 



HEAT ENGINES 



CHAP. 



Slide 
Valve 



Crank 




Piston 
Piston Rod 



Gudgeon Pin 



Wheel 



are further provided with a pump to remove the air and water pro- 
& bv steam on condensation. m 

Again in powerful engines the 
Steam Ch©sT_ high pressure steam is not allowed 

to expand completely in a single 
cylinder. The steam is partly ex- 
panded in one cylinder and passed 
on to one or more cylinders where 
the expansion is completed. Such 
engines are known as compound 
engines and may consist of three 
or four cylinders. 

Richard Trevithick was. the first 
to construct a 'locomotive 1 , i.e., a 
steam engine which con Id draw 
carriages on rails. He could 
however, push his inventions to 
financial success. It was left to 
George and Robert Stephenson, 
father and son, to construct the 
First successful locomotive — the 
"Rocket", and run the first rail- 
way train in 1829, between Liver- 
and Manchester. Robert 

on was the first to apply the 
steam ei Q ships in 181;:. 

12. Efficiency of Engines and Indicator Diagrams. — The earlier 

ventors of steam engine had no clear idea of the Nature of Heat, 
ra rather than physicists, they did not make seri- 
-. attempts at understanding the physi involved in the 

running of a steam engine. They measured efficiency by finding out 
the quantity of coal which had to be burnt per unit of time in order 
to develop a certain power. Tin's was rather a commercial way of 
measuring efficiei 

An absolute measurement of efficiency is obtained from the first 
law of thermodynamics. A heat engine is merely an apparatus for 
rsion of heat to work. THe heat supplied is obtained by 
finding out the calorific value of the fuel consumed by burning a 
sample of the fuel in a bomb calorimeter (p. 63). If Q be the calori- 
fic value of the fuel consumed per unit of time and W the power 
developed, we can define the economic efficiency t; as the ratio be- 
tween Q and W, viz. rj = W/JQ. Accordingly tj is the fraction of 
the heat, converted to work. 

For an ideal engine, ■>} should be unity. But actual experience 
shows that i) is rather a small fraction. In Watt's days, it was only 
h% ', now-a-days even in the best type of steam-engines, it hardly 
exceeds 17%. 

The question arises whether this lack of perfectness is to be 
ascribed to the bad designing of heat engines, or whether there is 



Shaft £ 

Eccentric 



.—Alain parts i 

;ine. 



THE CARNOT ENGINE 



213 



ix.l 

something in the very nature of things which prevents us from con- 
verting the whole amount of heat to work. 

This question was pondered over by Sadi Carnot about a hur> 
m -,,i years He showed that even with an ideal engine, it is 

■possible to convert more than a certain percentage of heat to 
work. 

It is very convenient to represent the behaviour of an engine 
by an indicator diagram and hence iu discussing the theory and per- 
formance of heat engines this is always 
done. Suppose a certain amount of 
gas is contained in a vessel at a certain 

perature and pressure and occupies 
a certain volume. Evidently the state 
of the substance is uniquely represent- 1> 
ed by assigning its pressure and volume. 
Thus we can represent the state of the 
gas by a point A (Fig. 9) on a graph 
such that the abscissa of the point re- 
presents the volume of the gas and the 

i :ate represents the pressure. 

Let the pressure and the volume 
of the gas be changed to that corres- 




Ffe. 9. Ti: 



ponding to the point B and suppose the pressure and the- volume 
i rnout this change are represented by points on the line All. 

Then this operation is represented by the line AB on this dur 

Such p-v diagrams arc known as indicator diagrams* 

\s proved on p, 4fi the work done by the gas in expanding against 

a pressure p is p*S. In this case since p changes from point to point 
ral work done by the gas in expanding from :•, to v* is equal 



to f v * pdv and is evidently equal to the area AabB, The work is 

taken to be positive if the diagram is traced in the clockwise direction. 
It is thus clear that the indicator diagram directly indicates the 
work done bv the engine during each cycle of operations, the work 
being equal to the area a by the indicator diagram, indicator 

diagrams are therefore of great use in engineering practice. 

The indicator diagram gives the work performed by the piston 
per stroke. Multiplying it by the number of strokes per second i.e. 
bv twice the number of revolutions of the shaft, we get the power 
indicated by the indicator which is generally expressed in horse-power 
_;,.,. p a The power actually delivered by the engine 

is measured by a brake dynamometer and is called the brake horse- 
wer, the ratio of this to the _ indicated horse-power is called the 
medio:; in; I efficiency of the engine. 

13. The Carnot Engine.— -As we have seen, the function of the 
steam engine is to convert the chemical energy stored in coal to 
energy of motion by utilising the expansive power of steam. 



214 



HEAT ENGINES 



[CHAP. 

The machinery necessary for this purpose is, however, so com- 
plicated that one is apt to lose sight of the essential physical prin- 
ciples in the details of mechanical construction. Lee us, therefore, 
discuss the physical principles involved in the running of a heat 
engine. Three things are apparently necessary, viz., a source of 
heat, a working substance, and machines. In the steam engine, the 
source of heat is the furnace where heat is supplied by the burning 
of coal. But we may get heat by a variety of other means, e.g. s 
by burning oil, wood, naphtha, or even directly from the sun (solar 
engines) , or from the inside of volcanoes (as is sometimes done in 
Italy). We can, therefore, replace the furnace by the general term 
"reservoir of heat/* For steam, we can use 'the general term 
1 working substance," for any substance which expands on heating 
can be used for driving heat engines. As a matter of fact, we have 
got hot air engines in which air is heated by a gas burner or a 
kerosene lamp, and pushes the piston up and down as steam would 
do. 

In addition to the three requisites mentioned above, we require 
a fourth onc,_ viz., the possibility of having a temperature difference. 
1 his at first^ is not so apparent, but can easily be made clear. In a 
hot air engine, the heated air can push the piston outward since 
the air outside is at a lower temperature. If there were no difference 
of temperature, no difference of pressure could be created, hence the 
machine would not work. We, therefore, requhe not only a sou 
of heat, but also a sink, i.e., a heat, reservoir at a lower temperature 
In .steam engines the surrounding air acts as the sink of heat or 
con i 

bserved that the fund the machine is to extract 

a certain quantity of heat Q horn the heat reservoir F, convert a 
part of it to work and transfer the rest to the heat sink G. He 

also showed how these operations 
should be carried out so that the 
efficiency may be maximum. 

Since whenever there is a differ- 
ence of temperature, there is a pos- 
sibility of converting heat to work, 
the converse is also true, i.e. if we 
allow heat to pass from F to G by 
conduction, wc miss our opportunity' 
of getting work. Hence we must 
extract heat from F in such a way 
that loss of heat by conduction is 
reduced to a minimum. Carnot, 
m« m ti - a 1 ^ lefore, thought of the following 

s ' 10 :~V^. ld f ' £ arnot en «'» e ideal arrangement, 
with indicator diagram. t 5 

^ rr,. ,,-s, * F T ' s a Jieat reservoir at tempera- 

ture T (Fig. 10) , G a heat-sink at T f , S is the cylinder of the engine 
containing a perfect gas instead of steam as the working substance 
and fitted with a non-conducting piston. The walls 'of the cylinder 







EL] 



THE CARNOT ENGINE 



215 



are impervious to heat but the bottom is perfectly conducting. The 
behaviour of the working gas is shown by the indicator diagram 
showing the pressure and the volume of the gas at any instant. i>et 
the following steps be performed :— 

(1) Let the initial temperature of the gas within S be T and let 
it be pitted in contact with F, and die piston moved forward slowly 
Ah the piste, : - the temperature tends to fall, and heat will pa^ 
from F to S. The operation is performed very slowly, so that the 
temperature ol the gas is always 7\ The. representative point on the 
indicator diagram moves from A to B along an isothermal curve. 1 he 
heat O cted in this process is equal to the work done by the 
piston in free expansion, and is given by 

= = f B pdv = RT log, £ - area AabB. . . (1) 

(2) F is then removed and H, which is simply a non-conducting 
cap, is applied to the cylinder, and the piston allowed to move for- 

d (by inertia). Then the *as will describe the atflabatic BC and 
will faU ill temperature. We stop at C when the temperature has 
fallen to P. 'I he work done by the gas is given by 

where piP — K"=£ & o v = AbV< 

Since the pressure is now very much diminished, die gas has 
lost its expansive power, hence in order to enable it to recover its 
capacity Tor doing work it must be brought back to its original con- 
dition. To effect this we compress the gas m two stages: first, 
isothermalh all ng the path CD, and then adiabaticaUy along DA. 
The point D is obtained by drawing the isothermal T through U 
and the adiabatic through A. 

(B) During the isothermal compression, the cylinder is placed 
in contact with the sink G at !P. The heat which i s developed 
owing to compression will now pass to the sink. This is equal to 
the work done on the gas and is equal to 

= area CcdD. . (3) 



-. 0' = \°pd»=* tfT'log,-^ 



(4) The cylinder S is now placed in contact with II and the 
gas is compressed adiahatically. The work done on the gas by adia- 
batic compression is 

Wi = f "/wfc = ~( T- T') - area DdaA. . 
It is thus seen that m z t=W4, 



m 



210 



HEAT ENGINES 



The net work done by the engine 

W — w x -\. w s w 4 — area A1JCD, 



[CHAP. 

(6) 



The last result can be written down directly from the first law 
of thermodynamics. 

Since B and C Ik on the same ad i aba tic, we have by equation 
24, p. 48., 1 

■:'...-■" ■'">■.- - 1 

(£P" <?> 

— I ■/-• J — Pi the adiabatic expansion ratio. 



or 



Simi • 






n 



prf 



or 



= :r^= r .» the isothermal expansion ratio 
have, therefore, 

Q = RT log r, £' = i2:r log r 

and Tr-0-/>' = i;.;;r-T') log • r. . 

r 3"' ""■•/-: 



Hence 






-afV'j 



(S) 
(9) 



of the Cartiot en: 

W T' 

-HIP < 12) 

analysed the moi ion of the Camot cycle we now 

I to .show that (I) it is reversible at cadi stage. (2) that no 
engine can be more efficient than the Carnot engine, m that the 
re of the working substance is immaterial. 

14. Reversible and Irreversible Processes,— A process is 

one which can be retraced in the opposite direction to that h 
substance passes through exactly the same states in all t8 as 

in the direct process. Further, the thermal and mechanical effects 
at each stage should be exactly reversed, U., the amounts of heat 
ived and of work done in each step should be the same as in the 
direct process but with opposite sign. That is, where heat is 
absorbed in the direct process it should be given out, in the reverse 
cess and ; : -, th and where work is done bv the working sub- 

stance m die direct process, an equal amount of work should be 



DC] 



RSIBLE AND IRREVERSIBLE PROCESSES 



21? 



done on the working substance in the reverse process. Processes in 
which this does not take place are called irreversible. 

For clarity we may add some examples of reversj 

F 1 '■■••-■'- • - transfer of heat from one bodv to another can be 

reversible only when the two bodies are at the same temperature. 
In case of two bodies at different temperatures, the transfer of 
heat occurring by conduction or radiation cannot be reversed and 
the process is irn ible. 

We shall now consider examples of reversible processes. It is 
r from the above definition that the process of bringing an elastic 
substance into a definite state of stress very slowly is revers 
because for a given strain the substance has always a' definite stc 
A convenient mechanical example of a reversible pro* afforded 

by the performance of a spring balance in the following way; When 
the spring is very slowly stretched work is done upon it. If, on the 
other hand, it is allowed to contract slowly bv the same amount 
the same amount of work is done by the spring, for die work done 
in increasing the length of the spring by Si is equal to the product 
of the force F and U. Both these quantities depend upon the si 
of the spring at the instant and ave the same value whet] 

l f c s P n ' : : ding or contracting. Thus the work done upon 

n f.. be equal to the work done bv the ■ 

wlu - the reverse process and the process 

[ f n ■'■• ■■'< however, that the stretching must be 

d or reduced gradually by the c 'ion of a- force 

• mftnitesimaUy at every instant, from the stn 
developed m the spri ,, part of the work will be spent 

in setting up vibrations of the spring and this will 
ability Such a process is called a quasi-static process and consists 
essentially of a ,. m of equilibrium states. 

The case of elastic fluids is analogous to that of the spring To- 
:v volume of the fluid there corresponds a definite stress 
pressure, so that the amounts of work done during a balanced expj 
sion or compression are equal. This is an important example o£ a 
reversible process. It is important to note that the expansion should 
be balanced otherwise whirls and eddies may he set up in the fluid 
which will gradually subside on account of flui< i :i ion with the pro- 
duction of heal and thus a part of the mechanical work would be 
lost. Such expansion or contraction may be either isothermal or 
acLxababc and can be brought about easily bv applying pressure on 
piston enclosing the fluid and adjusting the pressure to differ from 
the fluid pressure by an infinitesimal amount. 

Examples of irreversible processes are (I) sudden unbalanced 
expansion of a gas, either isothermal or adiabatic, (2) TouIcThotri- 
expansion, (3) heat produced by friction, (4) heat generated when a 
i ent flows through an electrical resistance, (5) 'exchange of heat 
5? 1 "52! ! ? odles ,^ differ e nt temperatures by conduction or radiation 
(n) diffusion of liquids or gases etc. Examples (1) and (2) exhibit 
internal or external mechanical irreversibility, (4) and (5)' exhibit 
thermal irreversibility, and (6) exhibits chemical irreversibility. 



21 a 



HEAT ENCINF5 



[CHAP- 



A reversible process may be represented by a line on the indicator 
diagram (p, v) out an irreversible transformation cannot be so 
represented. 

14. Reversibility of the Carnot Cycle*— It is now important to 
notice that the Carnot cycle is reversible at each stage, i.e., instead 
of abstracting the heat Q from a source T and transferring a part Q' 
i sink T' and converting the balance Q - Q' to work, we can proceed 
in such a way that the machine abstracts the heat <T from the sink 
at T, then we perform the work W on the machine, and Q is trans- 
ferred to I source at 7\ This is done by proceeding along the 
reverse route ADCBA, i.e., first allowing the gas to expand adiabati- 
cally from A to D, then allowing it to expand isothermally from D 
to C in contact with the sink at V, the heat (T being extracted in 
the process. Then we compress the gas adiabatically till we reach 
the point B (temperature T) . Next S is placed in contact with the 
source, the gas is further compressed isothermally till we reach A, and 

heat (Ms transferred to the body at the higher temperature T. 
The machine, therefore, acts as a refrigerator, :>,, by performing 
the work W on it, we are depriving a colder body T* of the heat O . 

An engine in which the working substance performs a reversible 
cycle is called a reversible engine. Engines in which the cycle is 

versible are called irreversible engines. Fur the Carnot cycle to 

be reversible it is essential that the working substance should not 

differ sei erature from that of the hot body and the 

condensi l it is exchanging heat with them. There should be 

of heat by conduction in the usual sense, for it. would be 

[b] . This ires that the isothermal processes AR, CD 

bed indefinitely slowly, and the source and the con- 

arge capacity for heat so as not to 

change in temperature during the proot 

ain the piston should move very slowly without friction. The 
changes in volume should be brought about by very small changes in 
the load on the piston in both the .isothermal and adiabatic processes, 
so that the difference between the external pressure and the pressure 
of the gas should always be infinitesimally small. Tims the Carnot 
cycle postulates the existence of stationary states of equilibrium 
while in an actual process the physical state is always changing. 
Further there should be no loss of heat by conduction from the 
to the piston and cylinder. Tt will thus be seen that the Carnot cycle 
with its perfect reversibility is only ideal and cannot be realised in 
practice, 

Nevertheless, it should be noted that for theoretical purposes die 

deviations from the ideal state may be lightly neglected if they are 

not an essential feature of the process and if they can be diminished 

much as may be desired by suitable devices and then corrected for. 

16, Carnofs Theorem.— The idea of reversibility is of the great- 
est importance in thermodynamics for the reason that, working 
between the same initial and final temperatures, no engine can I 






IX.] 



CARNOT S THEOREM 



219 



■ent than a reversible engine. This is known as Garnet's theorem. 
We now proceed to prove this important theorem, 

Suppose we have two engines R and S, of which R is a reversible 

engine and S an irreversible engine. If possible, let S be n 

efficient than "R, Suppose S absorbs the heat £) from A, converts 

l W f to work and returns the rest, viz., Q~W to the condenser. 

Let S be coupled to R (Fig. 11) and be used to drive R backwards. 

We are using R as a refrigerator. It thus abstracts a certain 
amount of heat from B, has the 



work W performed on it, and 

returns the same heat Q to the 
source A. The amount "of heat 
that R abstracts from B must 
equal Q-W* Now since S is 
assumed to be more efficient than 
R, W>W and hence Q— FF> 
Q - W r , vi&, R abstracts more 
heat Iron: B than S restores to 
it. Thus the net result is that the 
compound engine RS abstracts 
heat (W-W) per cycle from E 
and converts the whole of it to 
work,f while the source is un- 



N 



/* 



Hot Bod 1 ? 
A 



> 



v R ^ 



ill 



•-.-., 



y 



B 

Q-W' 



\ 



• II,— Coupling a reversible and 

an irreversible engine. 



affected. We are thus enabled, by a set of machines, to deprive a 
body continuously of its heat-content and convert the whole of it 
work without producing any change in other bodies. The machine 
would tii us work simultaneously as a motor and a refrigerator and 
would be the most advantageous in die world. It does not violate 
the first law for we are creating energy out of heat. 

Impossibility of perpetual motion of the second type. But still 

the process is quite as good as perpetual motion! of "the first kind. 
for heat, is available to us in unlimited amount in' the atmosphere, in 
the soil or the ocean, and if the process were feasible, it would give 
us all essential advantages of a perpetual motion machine, viz., that 
of getting work without any expenditure, though not without enere 
Human experience forbids us to accept such a conclusion^ Hence 
we conclude that no engine can be more efficient than a reversible 
engine. 

Again, if we assume that a reversible engine using a particular 

* Reproduced from Ewing's Steam Engine by the kind permission of Messrs 
Maemil3a.ni & Co. 

t X!l'- £ work ^' ~ ^ which is available, mav he used in driving a motor. 

J this was a term in use amongst the medieval philosophers who. thought that 

a machine might he invented some day which will create work out of nothing 

for some mechanical contrivances which , v c-re intended to produce perpetual 

motion see Aml-ade, fiuw, 3 «gc 14. The gradual evolution of the Law of 

ration of Energy lead; : to the first law of thermodynamics showed that 

is purely chimerical, since energy can never be created out of nothing 

;'," V 11 -''' ^ transformed from one form to the other. 

1 1 Tins result of human experience is the fundamental basis of the Second 
Law of Thermodynamics. In fact, Kelvin stated the second Jaw in this form 
t*or fuller discussion, set Chap. X. 



220 



HEAT liNGINE* 



[CHAP, 



working substance is more efficient than another working with a differ- 
,1,1 substance, we arrive by a similar argument at the aame absurd 
result. Hence the efficiency is the same for all reversible engines 
this is the highest limit ; efficiency of any engine that can 

be constructed or imagined. This is Camot's theorem. Thus rever- 
sibility is the criterion of perfection in a heat-engine. Hence we sec 
that the efficiency of a reversible engine is maximum and is lnde 
pendent of the nature of the ing substance. We chose perfect 

as our working substance, since its equation of state being known, 
we can easily evaluate 17. 

Note : — It may be mentioned that Carnot was ignorant of the 
true nature of heat at the time when he published his theorem of 
efficiency- He followed the old caloric theory in his speculations 
according to which the quantity of caloric O contained in a gubsta 
was invariable. It was Clapeyron who showed that Camot's argu- 
ments and result* remained intact when the Kinetic Theory of Heat 
was introduced. 

17, Rankice's Cycle. — In the steam engine, as we saw in the 
foregoing pages, the working substance is a mixture of water arid water 
vapour. ° If, 'however, we perform the same Carnot cycle with this 
fluid as with perfect gas the efficiency would also be the same {Camot's 
theorem) . But the Carnot cycle was performed with the working sub- 
stance always in the same cylinder. We have already seen that with 
a xi vent, the je of heat consequent on alternate heal 

and of the cylinder the modem engine-; are provided v. 

Still, howevi work- 

ice were to perfon e the efficiency wouldbe 

with the organs so separated the adiabatic corny 
working substance in the last stride of the cycle becomes 
imp] : e . Hence the cycle is modified into what is known as 

- cycle. 
The Rankine cycle is represented in Fig. 12. A'B represents the 
conversion of water into steam in the boiler at temperature F 3 and 
e pi and its admission into the cylinder, EC the adiabatic 
"on in the cylinder, CD the transfer of steam from the cylinder 
t:r at T 3 and pa and its condensation, and DA the 
transfer by a separate feed-pump to the boiler. This separate feed- 
pump transfers the condensed water at D 
at T 2 and p s to the boiler and the pressure 
is consequently raised From p% to p x . At 
A the water is heated from T 2 to T 2 in 
the boiler and the cycle begins afresh. 
The indicator diagram for an engine per- 
forming the ideal Rankine cyi repre- 
sented by ABCD and this also represents 
the work done by the engine. The area 
Fig. 1 -The Rankine cycle. FADE lily called the Jecd-puinp term, 

is extremely small and is generally 
lected. The work done by the engine may, therefore, " be put equal 




IX,] 



TOTAL HEAT OF STEAM 



221 



to the area. FBCE. This area can be calculated in much the same- 
way as on p. 215 if we know the equation of state of steam. This 
is cumbrous and in engineering practice a simple procedure is adopted. 

Let us introduce the total heat function H = U ■{• pV. Then 

dll = dU + pdV -I- Vdp — Vdp 

for an adiabatic process, since in this case (Chap. II, §23) 

\.pdv=a. 

Integrating between the limits B and C (Fig. 12) we get 

ight-hand side represents the area FBCE. Hence the work done 
in Rankine's cycle per gram-molecule of steam is approximately equal 
to the heat-drop H n —He- This will hold whether the steam is super- 
heated, saturated or wet. 

The heat taken in by the working substance is that required to 
convert water at p y and T z into steam at p t and TV This is equal to 

H B - [H D | (/>i • - p^ V w \-H n II n approx. 
where 11%, Hn denote the total heat of steam at pi, T } and of water 
at y. , -lively, V w the volume of v*»ter at D. The term 

(pi — pi) V, v 1 nts the area FADE. The efficiency is 

Hs — Hq 

The values of the total heat function are readily obtained from steam 
tables or charts which have been prepared for engineering work as 
explained in § IS. 

It will be v seen that the efficiency in Rankine's cycle is less than 
in Carnot cycle for in the former some heat (viz., that required to heat 
the feed-water in the boiler from T 2 to T 7 , is taken at a lower tem- 
perature. In actual steam engines the efficiency is only about 
60 — 70% of the Rankine ideal and their indicator diagram resembles 
Fig. 12 with the corners rounded off. 

18, Total heat of steam. — For calculating numerical values* it 
is usual to assume that the total heat or enthalpy of saturated water 
. water in equilibrium, with i' : ; saturated vapour) at 0*C is ;••: . 
and calculate the changes in total heat. Then the total heat of steam 
at (9°C is defined as the amount of heat required to convert 1 gram 
of water at 0°C into steam at &°C Regnault gave for the total heat 
of steam H the following empirical formula 

H = 606,5 + 0.305 $ 

where denotes the boiling point. The total heat may be computed 
as follows : 

Let us raise the temperature of 1 gram of water from 0°C to its 



* Si.::' Keesom and 
data on .-"• ■ .' cmd solid 
■X 1936). 



ICeyes, Thermodynamic Properties of Steam, inch 
lolid phase, (Wiley, 1936), or Calletiiiar, Steam 






222 



HEAT F.NGTNES 



[CUA!\ 



boiling point (t°C) , Denoting the average specific heat at constani 
pressure by c, the heat required for this process will be U if we 

neglect* the small work done due to pressure changes. Thus the total 
of water at t°C is H. Now let us evaporate this liquid at its 
boiling point t°. Consider the state of affairs when q gram of vapour 
has been produced leaving (1-q) gram yet to be evaporated. Then 
q is called the dryness fraction. The volume of. vapour produced is 
qv and the increase in volume is q(v-iv) where v is the specific- 
volume of vapour at t°C and w the specific volume of water. Hence 
the external work done is pq (v-w) in mechanical units. Therefore 
the gain in total heat during evaporation is 

qL— qi -\- pq (v - to) 
where * h the increase in internal energy when unit muss of the liquid 
is evaporated. Hence the total heat of steam at dryness fraction q i 

H=ct-\- qL. 
For dry saturated steam (q =. I), this becomes 

B = St + L. 

If the dry saturated steam at t"C is further heated, its temperature 

rises and we now speak of it as superheated vapour. In the case of 

am the superheating is usually produced under such conditions that 

can assume that the superheating is carried out at constant pres- 

u i the temperature of the superheated vapour to be t t °C 

that there are (ti — i) degrees of superheat. If c/ he the mean 

ilir heat n\' the vapour at constant pressure, rhe total heat of super- 

H s ~-a{-L+c p '(t x - 

INTERNAL COMBUSTION ENGINESf 

19. Historical Introduction.— Like other types of heat engines 
the this type also dates front medieval limes, Ch. Huygens. 



* M ore rigorously 

(ill — d (« + h) — <*« + Pdv -f 



and 



(T< 



.'t> 



:QS\ 



-v H 



= Tds + vdp. 



+ Xr 



dH = 



-c 



dH 
dT 



)."+(X\ 



*--i;^fiMfr)> 



and we must integrate the right-hand side from 0"C (To'K) to the boiling point 
T and the pressure from the saturated vapour pressure ^,> at To to the saturated 
vapour pressure p at T, 

t These are called internal combustion engines because in them heat is. pro- 
by combustion of fuel inside the cylinder in contrast with the steam engine 
which may be called external combustion engine I cans h this tribe heat is 
produced in the boiler. The interna] combustion engine is much more efficient 
than the steam engine because the working substance can he heated to a much 
higher temperature (20OO°C). 



IX.] 



THE OTTO CYCLE 



%>$ 



the great Hutch physicist, proposed in 1680 an engine consisting oi 
a vertical cylinder and piston, in which the piston would be thrown 
upwards by the explosion ot a charge oL : gunpowder. This would I il 
the cylinder with hot gases which would eventually cool, and 
pis: Id be forced down by gravity. Each stroke would require 

a fresh charge of gunpowder. 

This engine was never used, probably owing to the difficulty oi 
introducing fuel alter every explosion but the idea pcrsisusd. The 
discovery o£ combustible gases (coal gas, producer gas...) and of mine- 
ral oils brought the idea within the range of practical possibilities, as 
the difficulty of supplying fuel promised solution. But many long: 
years of practical and theoretical study were necessary before such 
engines became commercially possible. 

7t is not possible to go into 
the details of these early at- 
tempts, and it will suffice to 
describe and explain the action 
of the two types* which have 
survived, viz., (I) the Otto 
engine in which heat is absorb- 
ed ai S) the 
'•in whij li heat is 
bed al constant pressure. 
ly BO per cent of" internal 
combustion engines today are 
of the Oi to type. We shall com- 
pare the action of these engine'- 
with that of the ideal Carnot 
engine in which heat is absorb- 
al constant temperature. 

20. The Otto Cycle,— The 

Ottof cycle we are now going 
Lo describe was originally pro- 
d by Beau de Rochas 
.1 ., but the practical diffi- 
culties were first overcome by 
L876. The working of 
iic. will be clear from 
I The engine consists of the cylinder and the piston, the 

cyl ider being provided with inlet valves for air and gas (if gas is 
. lor combustion) and exhaust, valves. The opening and closing 
ese valves are controlled by the motion of the piston. There 
our strokes in. a complete cycle: 

♦Sometimes internal combustion engines are divided: into (1) gas engine, 
(2) oil and petrol engines, ft is, however, more scientific to divide them with 
regard I they follow and not with regard to fuel. ' Hence ' 

have been divided into the constant volume, constant pressure and constant 
raturc type, 

t Nikolaus Oi I8M,), born at Schlangenbad in Germany, 3s best k: 

;:.-. the inventor of the four-stroke gas engine. 




Fig. 13. The four strokes of the Otto 

• ;.iue. 






224 HEAT ENGINES [CHAP. 

(i) The Charging Stroke. — In this the inlet valves are open, and 
a suitable mixture of air and gas is sucked into the . the 

forward m i of the piston. 

(if) The Compression Stroke^- During this stroke all the val 

are closed, and the combustible mixture is compressed adiabatics 
to about l/5ih of its original volume by the backward motion of I e 
ton. The temperature of the mixture is thereby raised to about 

; a 

At the end of the compression stroke, the mixture is fired by a 
series o£ sparks. 

(iii) The Working Stroke, -Hie piston is now thrown forward 
witl oree, since owing to combustion a large amount of beat 

developed which raises the temperature of the gas to Lboul 2000 D C 
and a corresponding high pressure is de\ 

(iv) The Scavenging or the Exhaust Stroke,— At the end of the 
third'. stroke, the cylinder is filled with a mixture of gases which is 
useless ; iher work. The exhaust valves are then opened, and 

die piston moves backward and forces the mixture taut. 
scav is complete, a fresh charge of gas and air is sucked in 

and a fresh cycle begins. 

The tli uamical behaviour is illustrated in the indicator 

■diagram (Fig. 1 

It should "■ inhered that air is the working substance in 

Otto engine, the function of gas or petrol being merely to heat 
the air by its combustion. EC represents 
the suction stroke (gas and air being sucked 
at atmospheric pressure) . CD represents 
the compression stroke. At D (pre 
about 5 atmospheres, temperature about 
G00 C C) the mixture is fired by a spark. A 
large amount of heat is liberated, and the 
representative point shifts to A (tempera- 
ture 2000 a C, pressure about 15 atmos- 
pheres), volume remaining constant, AB 
sents the working stroke. At B the 
ust valve is c and the pre; 

[alls to the atmospheric pressure GV t . 
is the scavenging stroke. 

The efficiency of the engine can be ea 
if'ed out. The amount of heat added 
b C a (T a —Tj) where C 9 is the : heat 

at constant volume. (We suppose that this quantity retains the 
same etween 6GQ°C, and 2000° C. which is only approximately 

true.) 

Heat rejected = C s ( T b - T r i . 

Hence efficiency tj=1--=t ,J' . • . (1$) 




Im.lt. 14 -I 

Otto cycle. 






t 



IX.] 



thk orro CYCLE 



225 



From the relation TV*-* = constant for an ad i aba tic (see p. 48, 
equation (24), we obtain 



T* iVt\*- 1 T d 



(14) 



where p is the adiabatic expansion ratio. Hence 

r.-r.np 



T B -T* 



Therefore the efficiency 



='-(1) 



I ',>-' 



- (15) 

We obtain a similar expression for efficiency in terms of the adia- 
compression ratio as we get m a Carnot cycle (see equation 12, 
p. 216). The question arises : why not try to make the Carnot cycle 
a practical possibility? ' 

We shall now treat this question in more detail. In desiei 
a machine, other factors have to be taken into consideration to 

Lion of theoretical efficiency. The engine must be, in 

the first, place, quick-acting, i.e., a cycle' ought to be completed as 

tunckly as possible. This practically rules out the Cajnot engine as 

Les up heat at constant temperature, the process is therefore 

and the engine becomes very bulky and heavy in relation 

ait. In the Otto engine, the absorption of heat is 

almost instantaneous. But even where quickness is not the deciding 

factor, there are other weighty arguments against the adoption of the 

Larnot cycle. Pressure inside the cylinder varies during a cycle and 

Lie machine must be so designed that it can withstand the maximum 

pressure. Hence practical considerations impose a second condition 

t.e the maximum pressure developed inside the cylinder durjW a 

cycle should not be too great. Further, we must have a reasonable 

amount of work per cycle. 

Now due to the adiabatic compression in the last stage of the 
Carnot cycle an enormous pressure is developed. A detailed consi- 
deration taking numerical values shows that working between the 
same two temperatures, say 2040°K and MCPK the Carnot engine 
ops a maximum pressure of about 1000 arm., while in the Otto 
engine it ; « only about 21 atm„ though of course, the efficiency is 
reduced from 83% to about 44%, The Carnot engine also rec 
a large volume for the cylinder. These considerations show clearly 
tAat Che Carnot engine is quite impracticable. It will have to be 
very bulky and very stout and the power output will be extremely 
spall compared to its bulk. 

In the case of the Diesel cycle which is described in the next 
section the maximum pressure developed is about 35 atm and the 
efficiency rises to about 55%, It is for this reason that the Diesel 
engine is employed in cases where we want a large output of work. 

15 



mi 



HEAT ENGINES 



[CHAP- 






DIESEL CYCLE 



227 



hnl KbMm 

D J A 



21. Diesel Cycle. — Diesel* was dissatisfied with the low efficiency 
of the Otto engine and began investigations with the idea that 
efficiency of the Carnot cycle may be reached by certain other contri- 
vances. ' He did not succeed in his attempt, but was Led to the 
invention o£ another engine which, for certai rposes, presents 

ked advantages over' the Otto engine, 
>w since 

it is easily seen that if the com- 
pression ratio p can be still further 
pushed up f ^ would substantially 
increase. But in the Otto ei 
p cannot be increased beyond a 
certain value (about 5) otherwise 
the mixture would be fired during 
pression before the spark 
passes. 

In the Diesel engine no gas or 
petrol is introduced during com- 
pression. The indicator diagram 
tor the cycle is shown in Fig. 15. 
In the first stage pure air is sucked 
in (EC in Fig.) an* I I to ah nut one seventeenth of its volume 

in fig.) . A valve is then opened, and oil or vapour is forced under 




IS.— The Diesel • 




Fig. 16.— Strokes in a Diesel Engine. 



(o) Beginning of suction stroke; air-valve open, 

i :'.',' i T: entitling of compression stroke; all valves closed. 

(c) BcRintiing of working stroke; oil valve open. 

(d) Working stroke in progress. Etui at End injection ; all valves closed, 
(j?) Beginning of scavenging stroke; exhaust valve open. 

♦Rudolf Diesel (1853-1913), barn in Paris of Get man parentage was the 
inventor of the heavy oil engine. He was engineer at Munich. 



ure. The oil burns spontaneously as the temperature in the 
is about 1000°C. and above the ignition point of the fueL 
>f oil is so regulated that during combustion, as the 
ptsinh forward, Lhe pressure remains constant (DA). 

. when the temperature has reached the maximum value, 

e supply of oil is cut off. The piston is allowed to 

', describing die adiabatic AB. At li a valve is opened, 

mill pressure drops to C. CE is the scavenging stroke in which the 

fixture is forced out, and the apparatus becomes ready 

ycle. 

cycle can be performed in a cylinder of the type shown 
16. The cylinder is provided with the air inlet, the oil supply 
md die exhaust valves. The action of the cycle will be very clearly 
folio wi ' 1 1 from the figure, 

instead of opening a valve at B we may allow the gas to expand 

F where the inside pressure has fallen to atmospheric pres- 

.iiii', and Lhe air mixed with the burnt gas be forced out by a back 

FE. As this would involve a rather large volume for the 

, this procedure is not adopted in actual practice. 

all now calculate the efficiency of the Diesel engine. Let 
m i"i calculate the efficiency for the imaginary cycle AF6D, 

Heat taken = C P ( T a — T d ). 

Heat rejected = C p {T f -T t ). 

relation p = cT v! ^ 1 ' 1 (equation 25, p. 48) we have 

<*-«* T d T a T m -T d 



m 






ii 






T f -T c 
f a -f d 

(y-l)!y 



T f -T c 



--(£:) 



v-i 



(16) 



(17) 



e for the efficiency' the same formula as for the Carnot 

tto cycle. But in the Diesel engine the cylinder is designed 

m pressure of about 34 atmospheres. This determines 

1 1 on ratio because p\fp t ■- p y =34. This gives p=12.6 

which is a substantial improvement on the OltO engine, 

ted above, we have to take the cycle DABG. Heat 
constant volume. Hence heat rejected is equal to 

c v (T b -r c ). 



" v " ~C fi (T A -T d ) 
Id h wer value of about 55$>. 



(18) 






HEAT ENGINES 



CHAP. 



The Diesel engine, therefore, consumes less End than the Otto 
engine, bun els it has to withstand a higher pressure it must be more 
robust. Also the mechanical difficulties are much greater, hut they 
have been successfully overcoi 

22. Seini-Dieset Engines.— We have seen that in Diesel engines 




To avoid these die Semi-Diesel engine, also called the hot-bulb <?.' 
has been invented, In this air is compressed to about 15 to 20 aim. 
instead o£ 35 atm. This air does not yet become hot enough to ignite 
the oil and is consequently passed through a hot bulb where it is 
heated by contact with the bulb to the required temperature. The 
hot bulb 'is simply a portion of the cylinder which is not cooled by the 
water jacket. Tt consequently becomes much heated and serves to 
heat the compressed air. Next the oil (fuel) is injected into this hot 
bulb where, by the combined action o£ the compressed air and the hot 
bulb, it ignites. For starting the machine the hot bulb must be 
heated by a separate flame. After a few cycles the bulb becomes 
hot and begins to function properly. Then the external flame is 
removed. The semi-Diesel engines now-a-days are generally two- 
stroke oil engines without automatic control valv 

23. The "National" Gas Engine,— We shall now describe a 
typical Otto engine. Fig. 17 shows a 100 horse-power "National" gas 
: ne. 




Fig. 17.— The "National" gas engine. 

The engine is a stationary horizontal one having a single cylinder 
A A. The piston PP is attached to the crank shaft by the connecting 
rod CC and (its gas-tight into the cylinder AA. H is the combustion 
chamber having the inlet valve 1, the gas valve G and the exhaust 
valve E. The cylinder, the combustion chamber and the valve casings 
are cooled bv the water jacket WW. L is the lubricator, V, V are 



rx,] 



STEAM TURBINES 



229 



plugs which may be removed in order to examine the valves. M is 
the ignition plug where a spark is produced by the "Magneto" and 
serves to ignite the charge. The various valves and the magneto are 
worked by suitable earns mounted on a shaft driven by the crank 
shaft. 

The working of the machine is identical with that indicated in 
Fig, 13. The machine is started by rotating the fly-wheel so 
the piston commences its outward stroke. An explosive mixture is 
introduced and ignited and the piston is thrown forward. Then the 
working follows as indicated on p. 223. 

24. Diesel Four-Stroke Engines.— - These engines employ a cylin- 
der and valves of the type shown in Fig. 16* In constructional de- 
sign they are not much different from the Otto engine. They are 
further provided with a high pressure air blast for injecting the liquid 
fuel. 

25. The Fuel*— Some engines employ gaseous fuels the chief of 
which are coal gas, producer gas, coke-oven gas, blast-furnace gas, 
water gas and natural gas (chiefly methane) . The common liquid 
fuels are petrol, kerosene oil, crude oil, benzol and alcohol. Tn the 
Otto i tigine, if liquid fuel is used, it must be first vaporised and then 

s generally employ crude oils and hence their 
fuel ' heap. 

26. Applications. — During the brief period of fifty years the 
internal combustion engines have been much developed and employed 
for a variety of purposes. The steam engine, we have seen, is very 
wasteful of fuel, but the internal combustion engine is much more 
economical and has consequently been widely employed. 

The Otto four-stroke engine is usually employed in motor cms 
and aeroplanes. Petrol vapour is used as fuel in both cases. Four- 
stroke Diesel engines are largely employed in driving ships. Where 
very large power is necessary, the two-stroke Diesels ate in common 
use. Diesel engine is more efficient and more economical of fuel. 
Still has combined a steam engine with a Diesel engine and has 
obtained much greater efficiency in steam locomotives (SLill-Kitson 
locomotives/'! 

STEAM TURBINES 

27. The Steam Tnrblne. — We shall now give a brief description 
of the engines belonging to the windmill type. Windmills are set in 
motion by the impact of wind on the vanes of the mill, and at first 
sight it may appear strange why they would be classed as heat 
engines. But a little reflection will show that the wind itself is due 
to the unequal heating of different parts of the earth's surface, hence 
the designation is justified. Etit wind-power is rather unreliable, 
hence steam turbines were invented in which the motive power is 
supplied by the artificial wind caused by high pressure steam. There 

two types of turbines :_ — (1) The Impulse Turbine in which steam 
issuing with great velocity from properly designed nozzles strikes 






HEAT ENGINES 



CHAP, 



against the blades set on a turbine wheel and in passing through them 
has its velocity altered in direction. This gives an impulse to the 
Wheel which is thereby set in rotation. (2) The Reaction Turbine 
is somewhat similar in principle to the Barker's Mill which, as a 
laboratory toy, has been known to generations of students. In this 
the wheel is set in rotation in the opposite direction by the reaction 
produced by a jet of steam issuing out of the blades set in the wheel. 
Though the steam turbines are simple in theory, in practice great 
mechanical and constructional difficulties had to be encountered, but 
thanks largely to the efforts of De Laval, Curtis and most of all to 
Sir Charles Parsons who may be described as the James Watt of the 
steam turbine, these difficulties were successfully overcome. To-day 
the turbine is a prime mover of great importance and is aim 
exdusivcly used as the source of power for all large power land 
stations for the generation of electricity and propulsion of big ships. 
Their great advantage lies in the fact that instead of the reciprocating 
motion of the ordinary steam engine, a uniform rotary motion is 
produced by a constant torque applied directly to the shaft. They are 
more efficient than other engines because there is no periodic change 
of temperature in any of their parts and hence no heat is wasted. 
The full expansive power of steam can be utilised under suitable con- 
ditions. 

28. The Theory of Steam Jet*.— The essential feature of the 
turbines is that, in them the heat energy of steam is first employed 
itself in motion inside the fixed nozzles or bladings 
(in reaction turbines) and this kinetic enej 
is utilised in doing work on the blades. The 
fixed nozzles are so designed as to convert 
most efficiently the thermal energy of steam 
into its kinetic energy and produce a con- 
stant jet. We shall, therefore, first study the 
formation of jet under these conditions. We 
shall derive an expression for the kinetic 
energy imparted to the steam in traversing 
the fixed nozzle from the region of high 
pressure B (Fig. 18) to the region of low pressure C. 

Let the pressure, volume, internal energy and velocity of steam 
in E per gram be denoted by p lt v lt ih and w ± and the correspond ins; 
i C by fa v., Us and w z . Then the gain in kinetic energy 
of steam per gram on traversing the nozzle is (u> 2 2 -n> x s )/2j cah 
The net work done on steam in this process as on p. 138 is p^> : — p$v s 
while the loss in internal energy is u t -u^. Thus on th nption 

that the process is adiabatic we have from the first law of thermo- 
dynamics, 

rjj {w^-w^) = p t Vj-; aa^Aj— k tt , . (19) 

i.e., the gain in kinetic energy is equal to the heat drop. Generally 
the initial velocity is sensibly zero and hence die kinetic energy of 
the issuing jet is given by the heat drop hi - h 2 . 




Fist. 18— The fixed 



IX.] 



THE DE LAVAL TURBINE 



281 




-£^~ m^E3* 



If we now assume that there is no loss of heat energy through 
friction or eddv currents in the nozzle, the heat drop At- feu equal 

to the heat drop from the pressure fa to p & under isen tropic condi- 
tions. This can be readilv obtained from steam tables and represents 
under most favourable condition the maximum work obtainable from 
the turbine. Under ideal conditions the ordinary steam engine would 
also yield the same amount of work, i.e. } equal to the heat drop. The 
neater efficiency of steam turbines is, however, mainly due to the 
following two causes:— (1) in the turbine diere are nowhere any 
periodic changes in temperature as are found in the cylinder of the 
steam engine ; (2) the turbine is capable of utilising low-pressure 
steam with full advantage. 

29. The De Laval Turbine.— The velocity which is theoretic! 
attainable in a steam jet is, therefore, enormous, and this causes 
extraordinary mechanical difficulties. 
For if we want to utilise the whole of 
this energy., by allowing the jet of 
steam to impinge on the blades or 
buckets of a turbine wheel, simple 
math: ! considerations show that 

the velocity of the blades should not 
be short of' one-half the velocity ol the 
or maximum efficiency we 
should have a speed of the order of 
100 metres/sec. at the periphery of the 
wheel which carries the blades. The 
number of revolutions per minute ; 
sary to produce such peripheral speeds 
for a wheel of moderate dimensions 
reaches the value of about 30,000 and 
such speeds are impracticable for two 
reasons. First, th no construc- 

tional materials which can withstand 
such enormous peripheral speeds. 

The rotating parts, would fly to pieces on account of the large 
centrifugal forces developed and there would be mechanical vibration 
of the rotating shaft. Secondly, if the shaft is to be coupled to 
another machine which should usually rotate at a much, lower speed, 
say 2000 r.p.m., the problems of gearing become almost insuper- 
able. These difficulties were first successfully overcome by De Laval 
for low-power turbines. For the gearing down, he used doublc-hel 

! • wheels with teeth of specially fine pitch. To enable the requisite 

peripheral speeds to be obtained' with safety, the turbine wheel was 

made very thick in the neighbourhood of the axis (see Fig. 19) so that 

it can withstand the stresses set up by rotation and further tile shaft 

made thin. 

A De Laval turbine is illustrated in Fig. 20. The steam jet 
issuing from the fixed nozzles impinges on the blades, has its direction 




Fig. 19.— Vertical section 
wheel of De Laval turbine. 



232 



HEAT ENGINES 



CHAP. 



changed by them and thereafter escapes to the exhaust. Due to this 

action of the steam the 
blades are set in rota- 
tion. These are known 
as 'impulse turbines' be- 
cause the steam in pass- 
ing through them is not 
accelerated but has only 
its direction changed. 
The interaction of steam 
with the blades is clear ly 
indicated in Fig. 21. In 
these turbines the bli 
are 'parallel*, i.e., a 
passage between two 
blades has nearly the 
same cross-sect 
everywhere so that the 
blades offer very little 
obstruction to the steam 
jet and hence the p 
sure of the steam re- 
mains practically the 
same on both sides • of 
the wheel. 




Fig. 20.- -A De Laval ti 



30. Division of tfie Pressure Drop into Stages. Rateau and 

Zoily Turbines. — As a prime mover, the De Laval turbine is quite 

efficient for producing 

genera- 

ed steam must be 
used so. that the j 
sure drop in the 
would be enormous, and 
ice for maximum effi- 
ciency the blades must 
move with very high 
velocity. .Such large 
velocities are incompati- 
ble with safety. For this 
reason, the entire pressure drop is divided into a number of stages. 
Each stage consists of one set of nozzles and one ring of moving 
blades mounted on a wheel fixed to the shaft. Across each - 
the pressure drop is small so that the blade velocity necessary for 
maximum efficiency is considerably reduced and can be easily attained 
in practice. This principle of pressure compounding was first given 
by Parsons and later adopted by Curtis, Rateau and Zolly to the De 
Laval type. Rateau turbine has 20 to 30 stages. Zolly has about 10. 




l r -!. 



21, — Action of steatn on the blade* 
a De Laval turbine. 



IX.] 



REACTION TUUB1N£S 



m 



Steam 



31. Reaction Turbines— Parsons' Work.— Parsons* began his re- 
searches on steam turbine about the same time as De Laval (1884) » 
but he did not accept the fetter's solution of the problem, and 
went on with his own investigations which were completed by 
about 1897. His researches resulted in the evolution of a completely 
; i — the so-called Reaction Turbines. The mechanism of this 
type is shown in Fig. 22* 

The essential feature of the Parsons' turbine is its blading 
system. This consists of alternate rings of a set of fixed blades 
mounted on the inside o£ the cylindrical ease or the stator, with a 
set of moving blades mounted on the shaft or the rotor, arranged 
between each pair of rings of fixed blades as shown in Fig-. 5 
There are no separate nozzles 
for producing a jet, but the 
fixed blades ace as nozzles. It 
will be noticed that the blades 
have convergent passages, i.e., 
the cross-section of the passage 
goes on decreasing towards die 
exit side. 

the s t e a m passes 

through the moving' blades, 

is a drop of pressure in 

equence of which the 
steam jet increases in velocity 
as it traverses the blade- 

age, and as it issues out, a 
backward reaction is produced 
on the blades which conse- 



Hill 



Fdhk 



Moving 

Btadea 

"Fixed 

Blades 



Moving 

Bkdes 



JJJJJ> 
JJJJJJ- 



Fifi. 22.- 



-Tbe blading- _ system 
M turbine. 



of 



quently begin to revolve. The 

steam then "again passes to the next set. of fixed blades where it again 

acquires velocity and enters the next set of moving blades nearly 

perpendicularly. 

The moving blades are secured in grooves on the rotor shaft 
with their lengths radially outwards and parallel to each other. The 
fixed blades are secured' in grooves in the enclosing- cylinder and 
project inwards almost touching the surface of the spindle. The 
steam is admitted parallel to the axis of the shaft (axial flow type) . 
One pair of rings of moving and fixed blades is said to form a stage, 
and .generally a turbine has a large number oE stages. 

Pressure Compounding. — On account of die peculiar form of the 
blade-passages, the pressure goes on falling rather slowly, so that 
hole pressure drop is spread over a large number of stages which 
may amount to 45 or even 100. In consequence of this arrangement 
the velocity of the jet is very small compared to that of the De 
Laval type and the requisite blade speed is 100 to 300 ft. per sec. 

* Charles Algernon Parsons (1854-1931) built the fir<L reaction turbine in 1886 
at Newcaslle-on-Tyne. He was awarded the Copley & the Rumford medals ot 
the Royal Society for hh invwitions. 



234 



HEAT ENGINES 



CHAP, 



This reduction of velocity brought about by the distribution of avail- 
able pressure over a large number of stages, constituted the master- 
invention of Parsons which was also applied to other turbines with 
great advantage as already mentioned. 

We have stated above that the steam enters the moving- blades 
perpendicularly. _ In actual practice, however, the steam enters the 
moving blades with some relative velocity giving it an impulse as in 
De Laval's turbine and so the turbine works partly by impulse and 
partly by reaction. Thus it will he seen that the distinction between 
'impulse' and 'reaction' turbines is more popular than scientific and 
should not be taken too literally. All turbines are driven by 'reac- 
tion/ due to the alteration of velocity of the steam jet in magnitude 
and direction. The real distinction between the two types is that in 
the 'impulse' turbines the reaction is due to the steam jet being 
slowed down while in the pure 'reaction' turbine the reaction is pro- 
duced by the acceleration imparted to the steam in the g blades 
themselves on account of the pressure drop. Still, however, by 
simply looking at the blades one can say whether the turbii 
the 'impulse* or 'reaction' type, the impulse blading being charaete 
rised by parallel passages and the reaction blading by the convergent 
passages. 

During the last thirty years high-pressure ga . i: been uti 

Used in place of steam to drive the blades of a turbine. In the inter. 

nal combustion gas turbine the combustion of some oil serves to heat 

sed air to a high temperature and high pressure, arid this 

high pressure gas drives the turbine wheel. 

32. Alternative types of Engines. — During the Second World 
War Chi developed in Germany and Italy 

mainly with the idea of devising a powerful means of propulsion for 
>speed aircraft. In a modern turbo-jet a mixture of air and fuel 
ticked in and compressed by a compressor and then ignited so as 
are. rhis high pressure gas traverses a ti 
bine where it gets accelerated and the pressure falls, the gas escap- 
ing at the rear o[ the machine with a high jet velocity, The machine 
on account of the reaction thus furnishedj move yard with tre- 

mendous velocity. In this way the modern jet plane has been able 
to attain supersonic velocities le< velocities greater than the velocity 
of sound. 

Mention may be made of the Rocket and the guided missile 
whose use by the Germans in World War II as V-2 rocket caught 
world imagination and whose potentialities for space travel are bou 
to make it an important tool of the future for scientific research. The 
essential difference between a rocket and a jet aft is that the 

Cornier carries its own oxygen supply and thus can operate in vacuum 
while the latter obtains oxygen from the air. The propellant. solid 
or liquid, ignites producing a jet and the rocket takes off. The 
final rocket velocity V r — zv log f M* where n> is tlje jet velocity 
and M* denotes the ratio of the initial mass of the rocket to the final 
mass after combustion. In the two-stage rocket a smaller second 



IX. 



TI HEMODYNAMICS OP REFRIGERATION 



235 




rocket is placed hi the nose which ignites a lew minutes after take- 
off and the second rocket detaches itself and forges ahead under its 
own power with a tremendous velocity. Such a multi-stag 
was used by the Russians to launch the first artificial earth sal 
on 4th October, 1957 and the first artificial planet on 2nd January, 
1959- The thrust developed by these rockets is enormous. 

THERMODYNAMICS OF REFRIGERATION 

33, Tn Chapter VI we described several refrigerating machines 

and considered the general principles upon which their action depends. 
We shall now investigate the problem of die efficiency of these 
machines. In order to compare the efficiency o£ the various machines 
we shall introduce a quantity called the coefficient of pe: nee of 

a refrigerating machine. If by the expenditure of work W a 
is deprived of its heat by an amount q, its coefficient of performance 
is given by q/W, 

Tn Section 15 we saw that if a Carnot engine with a perfect gas 
as the working substance is worked backwards between the source 
of heat at T] and the condenser at T.,, the working substance abs- 
tracts an amount of heat (X from the condenser and yields (h 
to the source where Q^ and Q* are connected by the relation 
Qt/Ti = Qg/Tjf The work done in driving the engine is Ql — Qz- 
Hence the coefficient of performance of a Carnot engine working 
refrigerating machine is 



a. 



r. 



di - Qa " ** - T f ' 



(20) 



It can now be easily proved much in the same way as on p. . 
that no refrigerating machine can have a coefficient of performance 
greater than that of a perfectly reversible machine working 
the same two temperatures, and that all reversible refrigerating 
machines working between the same temperatures have the same 
coefficient of performance. Tt thus follows that the coefficient of 
performance of any perfectly reversible refrigerating machine working 
between temperatures T 1 and T- 2 , whatever the nature of the working 
substance, is equal to T 2 / (7*i - T 2 ), and this is the maximum possible. 

A Carnot engine using perfect gas and working backwards will 
be a most efficient refrigerating machine but its output will be 
due to the small specil heat of gases. For large capacity we must 
employ a liquid of large latent heat of vaporization and allow it to 
evaporate at the lower temperature. We have seen in §6, p. 128 
how practical considerations have led us to select ammonia and a 
few other substances for this purpose. 

34. Efficiency of a Vapour Compression Machine.-— We shall 
now calculate the efficiency of the vapour compression machine 






HEAT ENGINES 



[OJAP, 




l\ ; 

E A ^T 



Fig. 23. — Cycle of a vapour 
compression machine. 



described on p> 127. Let us trace the cycle of changes which the 

working substance undergoes. For 
this purpose assume the valve V 
to be replaced by an expansion 
cylinder. AB (Fig, 23) represents 
the evaporation of the liquid in the 
refrigerator, BG the adiabatic 
compression of the vapour by the 
compressor, CD the liquefaction 
of the sufjsLance inside the con- 
denser and DA the adiabatic ex- 
pansion of the liquid in the expan- 
sion chamber (not shown) . The 
cycle ABCD is perfectly reversible 
and hence its coefficient of perfor- 
mance is given by equation (20) . 

In actual machines the throttle valve takes the place of the 
expansion q f Iinder* This is done for mechanical simplicity. Conse- 
quently the part DA of the ideal cycle is replaced by DEA, On 
account of die irreversible expansion through the valve there is a loss 
of efficiency. The losses are twofold. An amount of work equal to 
the area DEA is lost and there is also reduced refrigeration effect 
since some liquid evaporates before reaching the refrigerator. The 
coefficient of pei .: for the actual machine can be calculated as 

the case of Rank' le (n, 220) from the values of the total 

of the working sub; the different states. 

Books Recommended. 
i . \e, Engines. 

fug, Steam Engine and other Heat En 
Glazebrook. A Dictionary of Applied Physics, VoT. 1., Articles 
on Steam Engine ; Engines, Thermodynamics of Internal Combus- 
tion : Turbines., Development of ; Turbines, Physics of. 

4, Judge., Handbook of Aircraft Engines (1945) , Chapman & 
Hall Ltd. 

.1, Clarke, Interplanetary Flight (1952), Temple Press Ltd., 
London. 



CHAPTER X 



THERMODYNAMICS 

1. Scope of Thermodynamics. — ■ The object of the science of 
JTeinL is the stud) p£ all natural phenomena in which heat plays the 
leading part. We have hitherto studied a few of these phenomena 
in a detached way, viz./ properties of gases, change of state, varia- 
tion of heat, content, of a body (calorimetry) , There are other pheno- 
mena (e.g*., Radiarion, i.e., the production of light by heated bodies) 
which have not yet been treated ; there are others which, strictly 
speaking, do not come under physics, vis,. Chemical equilibria, Elec- 
tro-chemistry, but in which heat plays a very prominent part. The 
detailed study of all these phenomena is beyond the scope of this 
volume. The object of this chapter is simply to develo] 
methods for the study of all such phenomena. 

The first requisite for such studies is the definition of the thermal 
state of a body or system of bodies. Next comes the development of 
general principles* The studies fall under two heads-*- (a) the 
study of energy relationships, (it) the study of the direction in which 
changes take pi ace t The guiding" principles for the two cases are the 
First and the Second Laws of Thermodynamics. 

2. The Thermal State of a Body or System of Bodies.— The I 

mal state of a simple hoi ius body like a gas or a solid is 

defined by its temperature 7\ pressure p or the volume V, These 
quantities are, therefore, called Tkermodynamical Variables. Of the 
three quantities only two are independent, the third being automa- 
tically fixed by the value of the first two, for it is a common obser- 
vation that the pressure, volume or temperature of a substance in 
any state, solid, liquid or gas, is perfectly definite when the other 
two quantities are known. Hence for every substance the pressure, 
volume and temperature are connected by a relation of the form 

P = ffaT) (1) 

This is called the 'Equation of State' of the substance. The 
oE (1) is known only for a perfect gas (Chap. IV), But though (I) 
is erally unknown we can proceed a good deal towards study- 
ins the behaviour of a substance under different conditions by using 
differential expressions. In the subsequent work we shall occasionally 
use differential calculus and some of its important, theorems. The 
reader must clearly understand the physical interpretation of these 
and for that purpose the following- brief mathematical notes are 
appended, 

3. Mathematical Notes.— Suppose that y is some function of 
the variable x and is continuous over a certain range of x. Let 8x be 
some increment in the value of x and let Sy be the corresponding 



238 



THERMODYNAMICS 



[CHAP. 



Increment of y, such that y -j- &y still lies within the range of y con- 
tinuous* Then the value of the ratio £ as fix tends to zero is the 

ox 

differencial coefficient of y with respect to x and is written -r- . 

ax 

If x is a function of the two variables y and z., ue. t 

* = / <y, * 

there are two differential coefficients of the function. In the first 
case the function may be differentiated with respect to 5? keeping z 
constant, and in the second it may be differentiated with respect to 
ping y constant. These differential coefficients are known as 
parlial differential coefficients and are written 

».- I 

where the suffix outside the bracket denotes the variable which is 
kept constant during differentiation. 

In order to find the variation in the quantity x when y and z 
are both increased by the amounts Sy and &z we must make a double 
application of Taylor's theorem. Now if x=f(y) we have from 

or's theorem 



(3*1 _ 









When x is a function of the two variables y and z, the value of 
., z + &) may be obtained by first considering the second 
varj it z-f 8z and expanding in terms of Sy only 

teorem. In the second stage y is kept constant and the 
_ Ll - l) h are functions of z-f- Sz are expanded by Taylo >rem 

in Ll - rii Let us denote the partial derivative of f(j, z) with 

respect to v by /., and so on. Hence 

=f[J\ .--3:} H >, e+fc) 

KW- .,G?{.y 5 c-f-S^r) + higher terms in tj>. 

Expanding each term on the right-hand side by Taylor's theorem 
in terms of S2 we obtain 

+H<W/wCa *) +28>8c/, 4 o-, «) + (&e)v«t>!» z)} + . (3) 

If instead of expanding first in terms of Sv, we had expanded first 

m terms of 82 and then in terms of 8y we would have obtained an 

expression similar to (3) with the only difference that instead 

t y t iy t z) Id now obtain jL , (y t z)> Since these two values of 

Ar-f-Sx must be identical we have 



MATHEMATICAL NOTES 



or 



(*) 



i _ d~'.-: 

ie order in which the differentiation is performed is immab 
If the variations 8y, gz are continuously decreased so that the 
terms involving their higher powers may be neglected, equation (3) 
reduces to the form 

-By-/, +&£./( 



or 



&* 



S* 



is the theorem of partial differentiation. 
Let us consider the equation 



(5) 



(6) 



On comparison with (5) we see that if equation (6) is obtained from 
an equation of the form (2) we should have 



M 



!."-(&■ 



(7) 



Conversely, iF M and A r are of this form there must exist a functional 
relalion of the form (2) connecting the variables y and & When (2) 
exists condition (4) h satisfied which from (7) is equivalent to the 
condition 



(8) 



Bz 

Hence if condition (8) exists, equation (6) must result from an 
equation of the form (2) . Then Sx is said to be a perfect differential. 
The physical meaning of this is that the change in x for the change 
in the variables y and z from y L , z x to y 2 , z* is given by 

*i - x 2 = f(y t , Zt) - /(V 2 . ZsO , 
and is consequently independent of the path of transformation, 
depending only upon the initial and final values of the variables. 

There exist some physical processes for which a relation of the 
type (6) exists but (8) is not satisfied. Then §x is not a perfect 
liferent ial. As a mathematical example consider the expression 



2 



Then 



Sx « 3k dy ■ V 2fd& 
dx „ Bx a 

dj'dz dzdy 



m 



(10) 






-= 



240 



Tl I FRMODYNAMI CS 



[CHAP. 



e (9) cannot be integrated as can be easily verified, The condi- 
tion of integrability is just the reverse of (10) . If we multiply" (9) 
by yz the resulting function 3y 2 z*dy -j- 2y ? 'zdz = dfy^z 2 ) is integrable, 
The quantity yz is called the In --fating factor of the expression 

In equation (5), suppose z is kept constant, i,e. f §2 = 0. '1" 
on division by $x, we obtain 



or 



1 UAUd, 






which implies that the reciprocal of tile partial differential coefficient 
is equal to the inverted coefficient 

4. Some Physical Applications. — From equation (1) we have, on 
differentiation, in the general case 



*-(&),■*+{&).": 



(12) 



For an i. so baric or isopiestic process, dp = 0. Hence equation (12) 
yields 



dp_ 
dT 



) e U'/tU' 









> /a 



H 



,-r-) —coefficient of volume expansion in an iso- 
o 1 I p 

baric process (Chap. VII). 
v 1^7) = bulk modulus of elasticity in an isothermal 

process, or inverse of the coefficient of 

if nx:?. nihility. 

— l-^-l = coefficient, of pressure increase in. ar 

choric (constant volume) process. 
^Equation (13) shows that the volume elasticity, the volume co- 
efficient of expansion and the pressure coefficient of expansion are 
not independent but are inter-related. 

As an illustration, consider mercury at 0°C. and under atmos- 
pheric pressure. The coefficient of volume expansion is LSI >< JO -4 
per °C. and the coefficient of compressibility" per atmosphere is 
3.9 X 10-* Hence from (13) 



{ m 



1 Tdv ] 
v IdTl 



\of ! 



v l dp \ T 



cr = 4b -5 atm.. 



3»9xl0-< 



x.] 



DIFFERENT FORMS OF ENliRGY 



241 



and the pressure coefficient 

7fe*7." 



46-5. 



Hence it follows that a pressure of about 46 atm. is required to keep 
the volume of mercury constant when its temperature is raised from 
Q°C« to PC. Thus if the capillary tube of a mercury thermometer 
is just filled with mercury at 30 °G., the pressure exerted in the 
capillary when it is heated to 34 °G. will become 4 X 46\5 ^= 186 
atmospheres. 

5. Different Forms of Energy, — As thermodynamics is largely 
a study of energy-relationships, we must clearly understand the term 
'Energy'. The student must be familiar with the concept of 'Energy' 
from his study of mechanics. 'Energy' of a body may be denned as 
its capacity for doing work and is measured in ergs on the C.G.S. 
system of units. 

Everybody must be familiar with the two ordinary forms of energy, 
viz., the kinetic energy and the potential energy. A revolving fly- 
wheel possesses a large amount of kinetic energy which can be 
utilised in raising a piece of stone from the ground, Le,, in doing 
work. Tn this process the kinetic energy of the flywheel is converted 
into the potential energy of the mass of stone. The mass of stone 
also in its raised level possesses energy which we call potential 
energy and which can also be made to yield work. For when the 
stone falls down under gravity the flywheel can be coupled to a 
machine and be made Lo do work. Thus kinetic and potential ener- 
gies are mutually convertible and in a conservative system of forces 
the sum of the kinetic and potential energies is constant throughout 
the process. All moving bodies possess kinetic energy by virtue ol 
their motion and all bodies placed at a distance from a centre of 
force possess potential energy by virtue of their position. 

There are other forms of energy also with which the student ol 
physio must have become familiar, and which we shall now take 
up. We have already pointed out that heat consists in the motion 
of molecules and must therefore be a form of energy. The experi- 
ments of Joule and Rowland showed that there exists an exact pro- 
portionality between mechanical energy spent and the heat developed. 
Friction is the chief method of converting mechanical energy into 
heat. The conversion of heat into work has also been accomplished 
by the help of steam engines and other heat engines which have 
been completely described in Chap. TX. Thus it has been fully 
established that heat is a form of energy. 

^ Energy ma y be manifested in electrical phenomena also in a 
variety of ways. It is well known that when two current-carrying 
coils are placed with their planes parallel to each other, they attract 
or repel each other depending upon the direction of flow of the 
current. In this case we may take the more familiar example of the 
tramway. Here electric currents are made to run the carriage. Again 
16 



242 



THERMODYNAMICS 



[CHAP, 



-a wire carrying a current becomes heated Tints electricity is also a 
form of energy which can be easily converted into mechanical or 
thermal form." 

Energy may also be manifested as chemical energy. For we know 
that a bullet fired from a gun possesses enormous kinetic en< 
which is derived from the chemical actions that take place when the 
powder detonates. Further a large amount of heat is evolved during 
a number of chemical reactions, viz., in the combination of hydrogen 
and oxygen. Also the chemical aciions going on in a voltaic cell 
produce electrical current. Thus chemical energy can be easily con- 
ed into mechanical energy, heat, electrical energy, etc. 
There are various other forms of energy, all of which can be 
converted into some one of the other forms. The important ones are 
the following ;— magnetic energy, radiant energy (Chap XI) , surface 
energy (surface tension), radioactive energy and energy of gravitation. 
There are three important laws governing- energy transforma- 
tion :— (I) The Transmutation of Energy, (2) The Conservation of 
Energy, 0) The Degradation of Energy, 

0. The Transmutation of Energy. — This principle states that 
energv converted from one form to another, We have already- 

seen how mechanical energy, electrical energy and chemical energy 
can be converted into heat. In fact all kinds of energy can be directly 
.verted into heat. Mechanical energv obtained from a water-fall 
irking a turbine, may be utilised to generate electrical energy by 
Uphng the turbine to a dynamo. Electrical energv may be con- 
rted into thermal as in electrical furnaces, or into' chemical 

?rgy as in t: secondary cells. Heat energy can be 

ced into electrical energy as in the phenomena of thermo-electri- 
n to mechanical energy ; ces r and so on. Examples 

couM be multiplied indefinitely. 

The Conservation of Energy,— The principle states that during 
• interchanges of energy, the total energv of the 

tem remains constant. In other words energy is indestructible and 
unbeatable by any process. No formal proof of the law can be given. 
It is a generalisation based upon human experience and is amply 
home out by its consequences. The law arose out of the attempts of 
earlier philosophers to devise a machine which will do work without 
expenditure of energy. They however railed in their attempt- Later 
it was recognized that al! the machines invented were simplv devices 
to multiply force and could not multiply energy, and the true func- 
tion of n machine was to transmit work, Onlv that much work could 
he obtained from a machine as was supplied to it, and that too in 
idea cases. This shows that energy obeys the law of conservation. 
In these cases, however, all energy cannot be recovered as some is 
dissipated as heat But heat is also a form of energy, and on equating 
the work obtained plus the heat developed to the\vork done, we find 
that the relation of equality is satisfied. Thus the principle is estab- 

led for this case and follows from a result of human experience 
which may be stated m this form: 



x.] 



FIRST LAW OF THERMODYNAMICS 



243 



"It is impossible to design a m-it-chine which will create energy 
out of nothing and produce perpetual motion. Energy can only be 
transformed from one form to another" 

8. The Dissipation of Energy. — Energy, we have seen, is capable 
of existing in different forms. In some forms however it is more 
available to us than in other forms, Le t> we can get more work from 
energy in one form than in another. For example, mechanical energy 
is the highest available form while the heat energy of a body at a 
low temperature is in a much less available form. ' Now the law of 
degradation or dissipation of energy says that energy always tends to 
pass from a more available to a less available form. This is called 
the degradation or dissipation of energy, and is intimately connected 
with the Second Law of Thermodynamics ; it will therefore be taken 
up later. 

FIRST LAW OF THERMODYNAMICS 

9» The First Law* — We have alreadv enunciated the law of con- 
servation of energy. The First Law of Thermodynamics is simply a 
particular case of this general principle when one of the forms of 
energy is heat It may be mathematically stated in the form* 



sc> = w -f- m, 



(14) 



where SQ is the heat absorbed by the system, S£/ the increase in its 
internal energy and SW the amount of external work done by it. It 
may be remarked that 3Q is here expressed in energy units, and can be 
found by multiplying the heat added in calories by L the mechanical 
equivalent of heat. For a simple gas expanding against an external 
pressure p, SW = p8V, but for surfaces, magnetic bodies, etc, other 
suitable terms are to be added to take into account other forms of 
energy such as mechanical, electrical etc. The quantity 'internal 
energy' is defined with the help of the principle of conservation of 
energy by considering a thermally insulated system for which 
8U = -&W. Changes in internal energy can thus be measured in 
terms of the external work done on the system. 

Applications of the First Law 

10. Specific Heat of a Body. — It is easy to see that the internal 

energy of a body or a therm odynamical system depends entirely on 
its thermal state and is Uniquely given, by the independent thermo* 
dynamical coordinates. In the case of a simple homogeneous body 
we have seen on p. 237 that, any two of the variables p, v f T are 
sufficient to define its state uniquely. Choosing therefore V and T as 
independent variables we have 



U = f{V t T)> . 
* It was first stated in this form by Gausius. 



• OS) 



244 



THERMODYNAMICS 



[CHAP. 



where U and V refer to a gram-molecule of the substance. Differen- 
tiating (15) we get* 

f -(#).«■+ (w) r * ■ • • ™ 

For a perfect gas LI —0 (p. 47); for a gas obeying van der 
WaaJs* equation (see § 33) 



IBU\ _ a 

( 3 V I T F- • 



l£ an amount of heat 8.Q be added to a thermodynamical system* 
say a perfect gas, part of it goes to increase the internal energy of 
the gas by dV, while the remainder is spent in doing external work, 
e.g., when the gas expands by a volume dV against the pressure p. 
We have from the first law, 

8Q=idUi-pdV t , (17) 

where SQ, dU are expressed in energy units. Substituting for dU 
from (16) in (17), we get 

and dividing by dT we have 

Hl_ldU\ r idu\ ]dV 

SO 
Now ■ j^= C, the gram-molecular specific heat. If volume be 

kept constant, 

(#).-(#).-*■ • - - w 

from definition. If pressure be kept constant, I ~) =C b , the 

V dT f p 

specific heat at constant pressure. Hence 

For a perfect gas { ?£) ^ -0 and p (|^l) = E, hence using (18) 



*It is thus seen that the quantity dV is a [perfect differential of a function 
U at the variables dcfinmjf the slate of the bodv; we can therefore write dV 
place of SU. _ Tn contrast with this the quantities, SQ, 8lV are not perfect 
■ seen from the indicator diagrams since 



differentials. 



2 ."j 1 , 3 f" d^P*™ 1 "Pan the path of the transformation and not only upon 
the initial and Jmal states. Hence we have denoted these changes by $Q and &W 
Some people use 4 in place of these 3's hut then this d should "not" he taken 

It J e d ?ff crcntial scnsc - *" is however a perfect differential and we can write 
v\¥ = paV. 






WORK DONE IN CKKTAIN PROCESSES 



245 



C p -C; = R, ... * , (20) 

as has been deduced on p. '16. The relation for solids and liquids is 
complicated (see sec. 38, Example 1) . 

It is easy to see that 8Q, the heal added, cannot be deter- 
mined only from the inilial and final states of the body, and a 
ledge of how the heat, has been added is essential. Hence 
C, the specific heat, has no significance unless the external conditions 
are prescribed. We have defined C p and C , but there may be other 
processes, e.g., adiabatic, when §(> = and C = 0. 

11. Work done in Certain Processes. — The following expressions 
for work done in different processes have been already proved : — 

(a) Work done by a perfect gas in isothermal expansion is 

(vide p. 215) equal to 

RTlQg t (V & IV'.). 

(b) Work done by a perfect gas in adiabatic expansion in which 

the temperature falls from 7\ to T 2 (vide p. 215) 



y— 1 



(C) 



12. 



In the case of a cyclic process the substance returns to its 
initial state and hence dU = 0. Equation (17) then yields 
dQ_ = pdV/j, i.e., the heat absorbed by the substance is 
equal to die external work done by it' during the cyclic 
process. Thus in Carnot's cycle the work done during a 
cycle is Ql - (£., 

Discontinuous Changes in Energy — Latent Heat.— When a 

body is in the condensed state (solid or liquid) and is subjected to 
increasing temperature, the state may suddenly change from solid to 
liquid form (fusion), or from liquid to gaseous form (evaporation) , or 
from one crystalline form to another (allotropic modification) . In 
such discontinuous changes the energy also changes discontinuously 
(phenomena of latent heat). Let us consider the evaporation of 1 
gram of water at 100°G, and 1 atuL pressure. The application of 
First Law yields 



L = u 2 -u 1 -\-p{v z -v 1 ), 



(21) 



where %, v 2 respectively denote the internal .energy and volume of 
1 gram of vapour, and %, Vj_ the corresponding quantities for the 
liquid. 

Equation (21) expresses the fact that the heat added is spent in 
two stages : (1) in converting 1 gram of water to 1 gram of vapour 
having tile same volume, (2) in doing external work, whereby the 
vapour produced expands under constant pressure to its specific 
volume. 



246 



ISKRMODYNAMICS 



; CHAP. 



u^ — Ui is sometimes known as the internal latent heat and 
p(v s — Vj) as the external latent heat. For water at 100°C, 

Vi = 1 tc, v 2 = 1674 c.c. (specific volume of saturated vapour) , 

„ .. . 1-013 xlO»x 1673 . ..„ , 

Hence p(v t - v{\ — n-= — ^xi cal. sz 40.5 cal. 

1 N ' 4-lBx 10 7 

Since L = 538.7, the Internal Latent Heat « 2 - «i = 498.2 Cal, 

It is often convenient to denote the different states of aggregation 
of the same substance according to the following convention, [H 3 0], 
(H 2 0), H a O denote water in the solid, liquid and gaseous states 
respectively. The quantity taken is always a mol. unless otherwise 
stated. The symbolical equation 

H a O = (H a O) -J- 10,170 cal ... (22) 

expresses the experimentally observed fact that the energy content, 
of a mol. of H 2 (gas) at 0°G exceeds the energy-content of a mol 
o£ H.O (liquid) at'0°C. by 10,170 calories, die change taking place 
at constant volume. 
We have further 

(H 2 0) = [HaO] + 1^30 cal, . . (23) 

us, t energy of a mol. of liquid II s O at 0°C. exceeds the energy of a 

moL of ice at 0°C. by 1,430 cal 

From equations (22) and (23) we have by addition 

H s O = [H..O] + 11,600 cal. 

i.e., heat of sublimation of 1 mol. of ice at constant volume is 11,600 
calories. This is a consequence of the First Law applied to a physical 
process and can be easily verified, 

13. The First Law applied to Chemical Reaction* — Hess's Law of 
Constant Heat-sommatioiL-— The First Law can be extended to Chemi- 
cal Reactions, Thus when 2 mols, of H 2 and 1 moL of 2 are 
exploded together in a bomb calorimeter, 2 mols. of (H a O) are formed 
and 136,800 calories of heat are t-volved. Allowing [or the heat 
evolved in passing from the gaseous to the liquid state, we obtain 
from the First Law, since SQ = - 116,500 and BW =0, 

2t/ HQ + Po.-aOW.o = H6,500 cab 
!>.., the energy of 2 mols. of H-. and 1 mol. of Or, exceeds the energy 
of 2 mols, of 'H a O by 116,500 calories, 

Hess stated in 1840, before the First Law had been discov i : 
that if a reaction proceeds directly from state 1 Co state 2, and again 
through a seri: o£ intermediate states then the heat evolved in the 
direct change is equal to the algebraic sum of the heats of reaction 
in the intermediate stages. This is an important law in Thermo, 
chemistry and is known as Hess's Law of Constant Heat Summation. 

The law is illustrated by the following examples : — 

G (diamond) + O a = C0 2 -f 94,400 cal 
CO+ £O s = CO a + 68,300 caL 






x. 



SECOND LAW OF THERMODYNAMICS 



241 



We have by adding 

C (diamond) + 2 = CO s -J- 94,400 cal. 
This is verified by experiments. 

We take another example. Let the symbol (LiOH) m denote the 
solution : ol. of the substance. Then take the equations:— 

2 [Li] 4- 2 (H 2 0) = 2 (LiOH) an -f H s + 106,400 
2[LiH] -f 2 (H s O) = 2 (LiOH)*, -f 2H 2 + 63,200 
The energy evolved is very easily determined from solution ex- 
periments. Subtracting the lower equation from the upper one • 
obtain 

2[Lil -f H* = 2[LiH] +43/200 cal. 
i.e., the heat of formation of two mols of [LiH] from [Li] and T-l 2 
is 43,200 cal, 

THE SECOND LAW OF THERMODYNAMICS 
AND ENTROPY 

14. Scope of the Second Law.— The Second Law of Thermodyna- 
mics deals with a question which is not at all covered by the First 
Law, viz., the question of the direction in which an}' physical 
chemical process involving energy changes takes place. 

A few illustrations will clear the point. Let us consider tru 
thermodynamical process illustrated by the symbolic equation :— 

2H 2 -f- 2 = 2H 2 + U calories. 

The equation tells us that if 2 mols. of H 2 gas combine with o 
mol of O s , 2 mols of H.O vapour are formed and U calories of heat 
are evolved, or vice versa, when 2 mols of H 2 Q-vapour are decompa 
completely into H ; , and 2 , U calories of heat must, be a', irh 
This result is obtained as a matter of experience from ealorimetric 
experiments, and we interpret the result according to the First Law 
of Thermodynamics by saying that in the combination of 2 molecu 
of hydrogen with 1 molecule" of oxygen to form 2 molecules of H s O, 
[Tie diminution, in energy amounts to JU/N ergs. This can 

be further utilised in calculating energy relations in other reactions. 
But the first law cannot tell us in which direction the reaction will 
take place. If we have a mixture consisting of H 2J O a and H a O-vaponr 
in arbitrary proportions at some definite temperature and pressure, 
i lie first law cannot tell us whether some H 2 and Q 2 will combine to 
form H ; .0 or some H 2 will dissociate into H 2 and Q 2 , the system 
i hereby passing to a state of greater stability. 

Or we can take a physical example. When two bodies, A and B. 
exchange heat, the first law tells us that the heat Iosl by one body 
is equal to the heat gained by the other. But it does not tell us in 
which direction this heat will flow. Only experience tells us that 
heat will pass from the hotter body to the colder one spontaneously. 






248 



THlUtMODVNAMlCS 



[CHAP. 



but in speaking of hotter and colder bodies we are making use of a 
V physical concept (temperature) which is not at all included with- 
in die scope of the first law. Tor giving us guidance in deciding the 
question of direction in which a process will take place, we require 
a new principle and this principle, which arose out of Carnot's specu- 
lations about the convertibility of heat to work, is genially known 
as the Second Law of Thermodynamics. 

15. Preliminary Statement of the Second Law. — The problem of 

convertibility of. heat to work has been treated in Chap, IX. It is 
readily observed that the main question there was about the direction 
of energy transformation. Mechanical energy and heat are only 
different forms of energy, but while mechanical energy can be com- 
pletely converted to heat by such processes as friction, it is not possible 
to convert heat completely to work. Even by using reversible engines 
which are the most efficient, only a fraction can be converted to 
work. The conversion is therefore only partial. The question is, 
why is it so ? 

An answer to this question has been given already in Chap. IX, 
viz., if it were possible to design an engine more efficient than a 
reversible engine, we could continuously "convert heat to work, and 
this will produce perpetual motion of a kind. We may call (his 
perpetual motion of the second kind. We are convinced that this is 
not possible, and we may start from a statement which expresses 
our con vie Lion just as the first law is based on the conviction that 
Energy cannot be created, but can only be transformed. We may 
ateirfent In the following form: — 

. impossible to construct a Heat Engine which will con- 
ly abstract heat from a single body, mid 'convert the whole of 

:he working system," This 
ement is equivalent to Clausius or Lord Kelvin's" statement of die 
Second Law. 

Lord Kelvin stated the law in this, form : — '7* & impossible by 
/ inanimate material agency to derive mechanical effect from 
■ lion of matter by cooling it below the temperature of the 
lest of the surrounding objects," 

Clausius staled it in the form : — "It is impossible for a self-acting 
machine, unaided by any external agency, to convey heat from one 
body to another at a higher temperature,' or heat cannot of itself pass 
from a colder to a warmer body." 

It must be clearly understood that these statements refer To con- 
tinuous or cyclical processes. For it is always possible to ]et heat 
pass from a colder to a hotter body and at the same time the sub- 
stance may change to a state different from its in i tin! one. As an 
example consider an isolated system comprised of a cold compressed 
gas and a hot rarefied gas communicating with each other through a 
movable piston. When the piston is released, the hot vapour is com- 
pressed and heat is thereby transferred from the cold to the hot gas. 
The state of the two gases however changes as a result of the process. 



To face p. Z4S 




Lord Kelvin (p. 248) 

William Thomson, Lord Kelvin of Largs, born on June 26, 1824, in Belfast 

died qn December 17, 1907, His important contributions to Heat 

arc the Second Law, the absolute thermometric scale and 

the Joule-Kelvin effect. He carried on valuable 

researches in Electricitv. 










Rudolf Cuaustus (1822 (p. 248) 

at KosKn, he studied in Berlin and became Professor of Physics 
successively at Zurich, Wureburg and Bonn. Simultaneously with 
Lord Kelvin he announced tJie Second Law of Thermo- 
dynamics. He was one of the founders of the Kinetic 
Theory of Gases. 






X.] 



ABSOLUTE SCALE OF TEMPERATURE 



219 



16. Absolute Scale of Temperature.— Lord Kelvin showed in 1881 
that with the aid of the ideal Camot engine it was possible to define 
temperature in terms of energy, and the scale so obtained is inde- 
pendent of the nature of any particular substance, £ Kelvin started 
the ret.uk (p. 219) that' the efficiency of all reversible engines 
ing between two temperatures is a function of the two tem- 
peratures only and is independent of the nature of the working sub- 
stance. This may be mathematically stated in the form 

W 



Q* 



-/(*!> ^ 



(24) 



where W is the work done by the Camot engine, Qt the heat taken in 
and i3 6 2 tile temperatures* between which the engine works. 
Now W= Qi-C&f where Q* is the heat rejected. Hence 



^ 



=/&, *■). 



or 



ft. 



- F(e lt e t ), 



where F denotes some other function of L and Q%.„ 

It can easily shown that the ratio Qi/Qa can be expressed 
in the form «/f(0i)/^ (&i) ■ -"For if we have reversible engines working 
between the pairs of temperatures (0 a , d b ), {$$, 3 ), (B u 3 ) and the 
heat absorbed or evolved (as the ease may be) at these temperatures 
be <b> Gap gs, then 



^— m, 0*}, 



Q 



n. 



^ = FiB,, 6,), £- - F(0 1; 3 ). . (25) 



ft. ^ *" ft* < 

From these by multiplying the first two expressions we get 



(L 



= F(B lt fl B ) X F(B tl 3 ), 



Hence the function F must satisfy the relation 

F\9 X , 9 a ) = F(9 U 9 t ) X F%, 

This will be so only when F(Q it d- £ ) is of the form ^(0, )/${#, 
fore for any reversible engine 

ft* *&) " ' 

It will be seen that i| & 1 >tf 2( C>]>f.> 2 and therefore 'P(0 1 )>'ii(6 i ) ) 
i.e., i/>(0) is a quantity which increases monotonically with Q and may 
be used to measure temperature. Denote the value or ip (0) by t. 

The relation (27) then becomes 

0,1 *i 



(26) 

There- 



(27) 



& 



(28) 



* These may be measured on any arbitrary scale. 



250 



TJIERMODYNAM ICS 



CHAP, 



Equation (28) is used to define a new scale o£ temperature t which 
is tailed the absolute or the thermodynamic scale. This scale does 
not depend upon the properties of any particular substance for equa- 
tion (28) is universally true. The ratio of any two temperatures on 
this scale is equal to the ratio of the heat, taken in and neat rejected 
by an engine working reversibly between the two temperature. 1 ;. 

The zero of this scale (r=0) is that temperature at which 
(from 28) and hence Ti r — (^, Thus all the heat taken by 
the engine has been converted into'work and the efficiency of the 
engine is unity, t cannot be less than this, i.e., negative, for if it 
were so, Q 2 would be negative which implies that the engine would 
be drawing heat both from the source and the sink. This is impos- 
sible from the second law and hence t = is the lowest temperature 
conceivable. This is the absolute zero of temperature which need 
not be the temperature at which all molecular motions cease to exist 
as one might otherwise infer from elementary kinetic theory. 

The zero having been determined, let us fix the size of the degree.. 
In conformity with general usage we suppose that the interval be- 
tween the freezing and the boiling points of water is divided into 100 
equal parts on this scale, viz, f 



3*. 



h-ioo 



(29) 



The thermodynamic scale has thus been completely defined and fixed. 
scale can be actually realised in practice. 

For this we must investigate the property of some actual sub- 
stance. A perfect gas is very suitable for this' purpose. In fact we 
have already shown on p. 216 that for a Carnot engine using a perfect 
gas as the working substance, 






. (SO) 

where T lf T z are the temperatures measured on the perfect gas scale. 
Equation (28) combined with (30) yields 



(81) 



-*", 



'.:.. the ratios of any two temperatures on the thermodynamic 

and the perfect gas scale aTe equal. Thus the absolute 

narnic scale coincides with Che zero oC the per!) , ■ -■. ther- 
mometer; for if t o — O, r 2 = 0. Further from (29) and (31) since 
the Interval between the ice point and steam point is 100° on both 
scales, r, e , and T iM are identical, and in general from (31) other 
temperatures on the two scales are also identical, Thus the thermo- 
dynamic scale is given by the perfect gas thermometer,. For the 
methods of reducing the ordinary gas scale to perfect gas scale 
p. & 






x.) 



ENTROPY 

ENTROPY 



251 



17, Definition of Entropy.— The formulation of the Second Law, 
as on p. 248, is not of much use for finding out the direction of 
change in a chemical or physical process. We want a more general 
and mathematical formulation which was supplied by Clausius. 

Clausius shifted the interest from the problem of convertibility 
of heat to work, to that of change of thermal state of the working 
substance, for it is only the working substance which undergoes a 
thermodynamical change in the process. He observed that when the 
heat Q is added to it reversibly at temperature 7\ and £>' taken away 
from it reversibly at temperature T r we have 



a 

T 



a; -a 



(32) 




This property was utilised in defining an additional thermodyna- 
mical function (Entropy) for the gas which proved to be of great 
importance in the further development of thermodynamics. In the 
following deduction, we have followed Swing's treatment.* 

Let us take a gas in a definite state A (Fig. 1), and compare this 
state with another state B, which is obtained by the addition of heat 
energy, Jt is easy to see that the 
amount of heat added from outside may 
be anything, owing to the variety of 
ways in which heat can be added. This 
can be visualised by drawing curves be- 
twecn A and B. There may be an in- 
finite number of curves passing between 
A and B, corresponding to the infinite 
number of ways in which heat can be 
added and for all these the amount &Q 
of heat added will be different. It will 
be shown, however, that the integral 

measured along any reversible process does not depend on the path, 
but only on the coordinates of A and B. 

Let us draw two adi aha tics through A and B and draw an isother- 
mal DC at temperature t cutting these adiabatics in D and C respect- 
ively. Let us now divide the area A BCD into a number of elemeutai'V 
Carnot eyries #yyV as in Fig. 1 (xy, x'y r , isothermais, and xx', yf 

adiabatics). 

Let SQ be the amount of heat absorbed at temperature T while 
*Sce Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 271. 



--- --vC 



V 

Fig. 1. — The entropy 
function. 



252 



THERMODYNAMICS 



[CHAP. 



*] 



ENTROPY OF A SYSTEM 



253 



the point shifts from x to y'and Sq be the amount given up while it 
shifts from / to x'. Then we have 

HI 

7 t ' 

Adding up for all the Carnot cycles between A and B, we have 



M 



r -4-T* 



m 



This is the value of 






along- the zigzag path Aabxy, B, for 



the value of the integral along the adiabatics Aa, bx, ....is 0. 

In the limiting case when the Carnot cycles are infinitely small 
the zigzag path Aabx B coincides with AB, 



Hence 



I 



If-f=><- 



(34) 









r B so 

The value of die integral j -2?- is now seen to be equal to 

'XBq/t and hence it is independent of the path between A and B. 

Instead of taking the particular isothermal DC we may draw any 
other isothermal cutting AD and BC, the value of the integral cannot 
be thereby changed, for if we construct a number of Carnot cycles 
between the first and the second isothermals we have 

1 ' * I ... . 

- S8 ? = -, E&q*. 

The points A and 1 ! taken anywhere on the adiabatics AD. 

and BC respectively, the value of the integral will remain the same, 

for J ^ along an adiabadc is zero since S£> = Q. Hence it follows 

that if we pass from one adiabatic to another the expression I — =* 

J % T 

denotes a quantity which increases by a fixed amount independent of 

the manner of transformation. If we start from some zero state and 

denote the value of the integral f -£- by s A it follows that 

Jo ■* 

function of the coordinates of A alone. Clausius called it the entropy 
of the substance. Now since we are not familiar with the zero state 
of the body when it will be deprived of all its heat, we cannot, find 
in this way the absolute value of the entropy in any state. We arc- 
however generally concerned with changes in entropy and hence we 
can measure from any arbitrary zero as is done in the case of energy. 
To measure the entropy m any state we have to take the substance 
along any reversible process to its zero state and find the value of 
lAq/t along this process. The entropy, though conveniently meaaur- 



jj. is a 



ed with respect to a reversible process, has nothing to do with it and 
exists quite independently of it. 

The above treatment is perfectly general and w r ill hold whatever 
substance is taken as the working substance. Now if the work is 
done by expanding against an external pressure we have from the first 
law for any process reversible or irreversible, &Q = du -\- pdv, where p 
stands for the external pressure at every stage o[ the process. But 
for finding entropy we have to substitute the value of Sf> along a 
reversible process. Now the process is reversible only when "the 
external pressure p is always equal to the intrinsic pressure of the 
substance, i.e., when the expansion is balanced. Hence we get* 

J = j r J«+M' m 

where p denotes the pressure of the substance itself. 

It is easily seen that entropy is proportional to the mass of the 
substance taken, for if we take M grams of the substance dU — Mdu t 
dV =z Mdv and hence 5- Ms. 

Exercise. — Find the increase of entropy when 10 grams of ice 
at 0°C. melt and produce water at the same temperature, given that 
the latent heat of fusion of ice = SO calories/gm. 

Since the process is reversible and isothermal we have 
AS = i|£ 3 — =2.93 caL/degree C. 

18. Entropy of a System* — In the last section it was shown that 
if the entropy per unit mass of a substance is s the entropy of m 
grams of the substance is ms. Tt can be shown in a similar way that 
if we have a system of bodies in thermodynamical equilibrium with 
different thermodynamical variables and having masses m 1t nu. . and 
specific entropies s s , s 2 . . ., the entropy of the whole system "is 

S = »»!*! + m 2 s 2 + ... . . . . (30) 

19, Entropy remains Constant in Reversible Processes.— The sum of 

the entropies of all systems taking part in a reversible process remains 
constant. Considering the Carnot cycle we notice that the working 
substance has the same entropy at the end of the cycle as at the 
beginning since it returns to the same state. The loss in em 
of the source is Qi/Ti, the gain by the condenser is Qj/.T* and 
hence the net gain in the entropy of the system is 

T B -T L - U < 3 '> 

Thus the entropy of the whole system remains constant. 



* A more general definition is the following : — If from any cause whatever 
the unavailable energy of a system with reference to another system at TV under- 
goes an increase A£, then A-E/7V measures the increase of entropy of the 
system. 



254 



1H KRMODYN AM I CS 



[chap. 



In the isothermal expansion of the working substance in the first 
part of the Carnot cycle the increase in entropy of the working sub- 
stance is (li/T u the loss by the source is Ql/T\ and hence the total 
entropy remains constant. This holds for any reversible transforma- 
tion* 

Working out as on p. 215 we see that the increase in entropy 
of a gram-molecule of perfect gas in isothermal expansion from volume 
F 3 to V 2 at temperature T is 



but by 



The change in entropy during a reversible adiabatic compression 
or expansion is zero. For in such a process the external work 

done is I pdV and change in entropy is /\S = \ ^ — 

the condition of the process from the first law dU-\-pdV =0, hence 
the entropy remains constant in an adiabatic process. Adiabatic 
curves are therefore sometimes known as "Iscntropics." 

20. Entropy Increases in Irreversible Processes. — The entropy of a 

system increases in all irreversible processes. As examples of such 

processes we may mention the conduction or radiation of heat, the 

rushing of a gas into a vacuum, the inter-diffusion of two gases, the 

opment of heat by friction, flow of electricity in conductors, etc 

<vt the theorem here for a few cases. 

Entropy during Conduction or Radiation of Heat, 
— Suppose a small quantity of heat (? is conducted away from a body 
A at temperature T, ro another body B at temperature T^{TiyT 2 ) , 
and I'. so small that the temperatures Tj and T 2 of the bodies 

arc not appreciably altered in the process, then A loses entropy 

equal to , r ~ while B gains entropy by an amount J? 

gain in entropy of the two bodies is 



Thus the total 



**=**(-* -it) 



m 



which is positive since Ti>T 2 . Hence the entropy of the system as a 
whole increases during thermal conduction. Similarly in the case of 
radiation, if a quantity of heat SQ is radiated from the body A to the 
body B, the gain in entropy is given by the same expression. Thus 
increase of entropy is always produced by the equalisation of tem- 
perature. 

(ii) Gas rustling into a vacuum. — Consider a perfect gas in a 
vessel rushing into an evacuated vessel and let the whole system 
be isolated. Since the gas is perfect the temperature does not change 



* In denotes natural logarithm. 



X. 



ENTROPY OF PERFECT GAS 



255 



in the process. The gain in entropy may be obtained by finding the 

value of I -=■ along any reversible path connecting the initial and 

final states. The reversible process most convenient for this purpose 
is an isothermal expansion of the gas against a pressure which is 
.always just less than the pressure of the" gas. 



For a perfect gas dV at constant temperature is zero, 

gain in entropy 



A$=4rf pdV = Rln A 



Hence the 



(40) 



where P it F 2 denote the initial and final volumes of the gas. Hence 
the entropy of the system increases in this irreversible process. 

21. Tne Entropy of a Perfect Gas. — Let us take m grams of a 
perfect gas having die temperature T and occupying the volume v. 
The entropy ms is given by the relation 

7" du-r-pdv 

o r~~ * 

Since du = mc„dT, p=zmRT/Mv f we have 



ms — 



ms = m (c v In T 4- -rj In v) 4- const. 



. (41) 

We can also express wis in terms of pressure. From the relation 

R 

M 



tp - e w -r 



wet get ms = m(c p In T — -^ In p) -\- const. 



(42) 



Exercise. — Calculate the increase in entropy of two grams of 
oxygen when its temperature is raised from <J D to 100°C and its 
volume is also doubled,. 



&S =2x2-3026 



5-035 



log 



373 

273 
Second 



+ 



I 



log 2 =0-184 



cah 



32 "*" 273 ' 32 ° j deg 

22. General Statement of Second Law of Thermodynamics, — 'We 

have shown above for a few irreversible processes that increase of 
entropy takes place during such a process. This result is however 
very general and holds for any process occurring of itself. We shall 
not, attempt to prove here this general statement, but only enunciate 
it in the following words : — Every physical or chemical process in 
nature takes place in such a way that the sum of entropies of all bodies 
taking part in the process increases. In the limiting case of a rever- 
sible process the sum of entropies remains constant. This the most 
general statement of the Second Law of Thermodynamics and is iden- 



256 



THKRMGDYNAMICS 



[CHAP. 



*•] 



PHYSICAL OONCKPT OF ENTROPY 



257 






tical with die Principle of Increase of Entr* viz,, the entropy 

a system of bodies tends to increase in all processes occurring in 
nature> if we include in the system all bodies affected by the change. 
Clausius summed up the First Law by saying that the energy 
of the world remains constant and the Second Law by saying that 
the entropy of the world tends to a maximum* Though the terras 
'energy of the world' and 'entropy of the world' are rather vague, still, 
if properly interpreted, these two statements sum up the two laws- 
remarkably well. 

23. Supposed Violation of the Second Law.— Maxwell invented an 
ingenious contrivance which violates the Second Law as enunciated 
above. Following Boltzmann's idea, he imagined an extraordinary 
being who could discriminate between the individual molecules. Sup- 
pose such a creature, usually known as Maxwell's demon, stands at 
the gate in a partition separating two volumes of the same gas at the 
same temperature and pressure and, by opening and shutting die gate 
at the proper moment, allows only the faster moving molecules to 
enter one enclosure and the slower molecules to enter the other 
enclosure. The result will be that the gas in one enclosure will be at 
a higher temperature than in the other and the entropy of the whole 
system thereby decreases, though no work has been done. This 
apparently violates the Second Law, 

We observe, however, that to Maxwell's demon the gas does not 
appear as a homogeneous mass but as a system compound o[ discrete 
molecules. The entropy does not hold for individual mole- 

cules, but. is a statistical law and has no meaning unless we deal with 
matter in bulk, 

24, Entropy and Unavailable Energy.— Consider a source at 
temperature T-y and suppose it yields a quantity of heat (> to a work- 
ing substance. If the lowest available temperature for the condenser 
of the Carnot cycle is T n the amount of heat rejected by the Carnot 
engine working 'between T t and T is QT /T V The remainder, i.e, 

Q— Q=? has been converted into work. Thus the available energy 

i s Q\ * — J- ) ■ Now suppose a quantity of heat $() passes by conduction 

from a body at. temperature T 2 to another at temperature TV The un- 
available energy initially was (T iY /T.,)Hl and finally it was (T /r 3 )SQ. 

Hence the gain in unavailable energy is SQ \Y"" t) T °' ^° bemg 

the lowest available temperature. The increase in entropy of the 

system is, as we have already proved, equal to SQ Isr- y). Hence 

the increase of unavailable energy is equal to the increase of entropy 
multiplied by the lowest temperature available. Thus entropy is a 






measure of the unavailability of energy, and the law of increase of 
entropy implies that the available energy in the world tends to zero., 
Le<, energy lends to pass from a more available form to a less available 
form. This is the law of degradation or dissipation of energy mention- 
ed on p. 243 and is thus seen to be equivalent to the Second Law* It 
follows as a corollary that all transformations, physical or chemical, 
involving changes of energy will cease when all the energy of the 
world is run down to its lowest form. 

25. Physical Concept of Entropy,-— The entropy of a substance 
is a real physical quantity which remains constant when the sub- 
stance undergoes a reversible adiabatic compression or expansion. It 
is a definite single-valued function of the Lhermodynaraical coordinates 
defining the state of the body, viz., the temperature, pressure, volume 
or internal energy as explained on p. 237. It is difficult to form a 
tangible conception of entropy because there is nothing physical to 
represent it ; it cannot be felt like temperature or pressure. It is a 

J statistical property of the system and is intimately connected with 
' the probability of that state. Growth of entropy implies a transition 
from a more to a less available energy, from a less probable to a more 
probable state, from an ordered to a less ordered state of affairs. The 
idea of entropy is necessitated by the existence of irreversible pro- 
cesses. It is measured in calories per degree or ergs per degree. 

26. Entropy-Temperature Diagrams. — We have represented the 
Carnot cycle and other cycles in Chap. IX by means of the usual in- 
dicator diagram in which the volume denotes the abscissa and the 
pressure denotes the ordinate. Another way to represent the cycle, 
which is often found very useful, is by plotting the temperature of the 
working substance as ordinate and the enixopy as abscissa. Its im- 
portance is readily perceived for during any reversible expansion the 
increase dS in entropy is given by 

dS = <f, 

where the integration is performed 
between the limits of the expansion. 
Thus the area of the curve on the 
: 1 1 iropy-leniperature diagram* represents 
the amount of heat taken up by the 
substance. 

The Carnot cycle can now be easily 
represented on the entropy-temperature 
diagram by Fig. 2. AR represents the 
isotlicrmal expansion at T\, BG the 
reversible adiabatic expansion, CD the isothermal compression at 
T 2 , the temperature of the sink, and DA the final adiabatic 
compression. It is evident that lines AB and CD will be 

* This is also called iephiyram as the symbol <f> was previously used for 
entropy, 

17 



D 



T* 



5! 



I* 



* Entrtfpy b 

Fsg._ 2,— Entropy-temperature 
diagram of Carnot cycle. 









258 



THERMODYNAMICS 



[CHAP. 



parallel to the entropy-axis and BC, AD will be parallel to the tem- 
perature-axis. The amount of heaL taken in is represented by the 
area ABba, heat rejected by the area DCbu, the difference ABCD 
being converted into work. These areas are respectively A (.$' - S), 
T 2 (& - 5) , (T t - T s ) (S f - S) and the efficiency is 1 - r 2 /7Y 

27. Entropy of Steam. — For enabling 1 the student to have a pro- 
per grasp of the conception of entropy we shall calculate the entropy 
o£ steam, Let us start with 1 gram of water at, 0°C. and go on adcf. 
trig heat to it. The increase in entropy when we add a small amount 

Of heat 8Q at T 6 is -p , But. SQ == vdT where o- is the specific heat 
■of water at constant pressure. Hence the entropy of water at T° is 
, cT ff dT . T 

if tr is assumed to be constant. s denotes the entropy of water at 
9°C) {T (1 D K.) and it is customary to put it equal to" zero. Hence 



for water 



- o log, --, 



This is represented bv llie 



where o- can he put equal to unity. 

logarithmic curve OA in Fig, 3. 

If the liquid is further heated it boils at the temperature T x ab 

its latent heat L. Since 
the temperature remains constant 
in this process the increase in entropy 
per gram is L/T v This is repre- 
sented by AB in the figure. Hence 
the entropy of 1 gram of dry satu- 
rated steam is 




If 






the steam is wet, q being its 
dryness, i.e., q is the proportion of 
dry steam in the mixture expressed as 
a fraction of the whole, the entropy 
of wet steam is 

Tf the steam is superheated to 

T.. c the entropy is further increased 



by the amount 



P 



dT 



where 



Entropy 



Fig. 3. — Entropy-temperature 
diagram o£ steam. 



denotes the specific heat of steam 
during superheating. This is repre- 
sented by BC in the figure. Hence 



x.l 



APPLICATION OF THE TWO LAWS 



259 



the entropy oL superheated steam at 7V 



= <? 1°S* 



n 



i- + r 



IS 



*P 



dT 

T 



As c* v 



273 ' T x j Tl 

,aries with temperature, its average value over small intervals 
this way the entropy of steam in any state can be 






is talten. In 
calculated. 

Exercise. — Find the entropy of saturated steam at a pressure of 
74 lb. per sq. in. 

From the table the boiling point at this pressure is 152.G°C. and 
the latent heat at this temperature is 503.6 cal./gm. The entropy 
of 1 gmt of water at 152,fi°G or 425.6°K is 1 X log, 425.6/273 = 0.44 
c&L/degree. The increase in entropy due to the evaporation at 
425!g°K is 503.6/425-6^ 1-18 cal./degree. Therefore the entropy of 
steam at 425.6°K is given bv .44 -f- 1,18 = 1.62 cal./degree. This may 
be verified by looking at the steam tables. For more accurate calcu- 
lation the variation of the specific heat of water with temperature 
should be taken into account. 

APPLICATIONS OF THE TWO LAWS OF 
THERMODYNAMICS 

28. General Consideration*.— For making frequent and effective 

use of the Second Law, it should be expressed in a convenient form. 
For this purpose a mathematical formulation is more desirable and 
was supplied by Clausius (p, 252) , was., 

&Q-r<fe* ( 4 *) 

where dQ is the heat absorbed during a reversible process connecting 
the two states. From the first law 

$Q=du + pdv, (7) 

where p denotes the intrinsic pressure of the substance and is given 
by its equation of state. Combining these two equations we get 

du + pdv=Tds (44) 

Let us further investigate the nature of the variations du, dv 
and <fo. As already stated in Chap. TV and also on p. 237, tire state 
of every substance is fixed if we know any two of the variables p, v, 
T, U....L For example, the internal energy uz=f(v, T) is uniquely 
defined it v and T are given. Thus if we represent the internal 
energy u of the substance" on the coordinate svstem v, T , the value 
of u for every point is fixed and does not depend upon the path along 
which we bring the substance to that point (state) ; in other words, 
the total change du has the same value whether we first make the 
change dv in v'and then the change dT in T or in the reverse order. 

* The results deduced from this equation in the following pages i and their 
confirmation by experiments may be taken as direct verification of this equation 

and therefore: of the Second ! ,r>\". 



260 



THERMODYNAMICS 



CHAl*. 



Mathematically, this means that du h a perfect differential and as 
explained on p. 239 we must have 



O'u 



a a — V7S (45) 

BxBy djdx 
where x and y are the two variables defining the state of the system. 
The entropy function was shown on p. 252 to be such that its 
value does not depend upon the path of transformation, i.e, f s is also 
a single-valued Junction of the coordinates. Similar is the case with 
the volume and the internal energy. 

As a contrast and for clearer understanding, it may be mentioned 
that pdv and lQ-= du -\- pdo are not perfect differentials for their 
values depend upon the path of transformation. This may be easily 

seen as T pdv represents th^ area of the indicator diagram and has 

different values for different paths. But though du -\- pdv is not in le- 
viable we have proved by physical arguments that (du -f- pdv) /T is 
in tegrable ; in other words, 1/7* is the integrating factor to the 
energy equation. 

We now proceed to make use of the above ideas for the further 
development of equation (44) . 

29. The Thermodynamkal Relationships (Maxwell).— Equation (44) 

may be written as 

du = Tds-pdv. .... (46) 

Now if x and y are any two independent variables we have, since «, 
s and v are expressible in terms of x and % 

, Bs Bs , 

j dv , , do , 
dv — ■=— ax- ■- ay, 
Bx By ■" 



and 



du = 



du , . Bit , 



Substituting the values of ds, dv, du in (46) and equating the co- 
efficients of dx t dy, we get 

<?«_ ~Bs _ dv 

Bx~ Bx ~ ** dx' 

Bv_ T ds _ dv 

"By 'dy p dy 

Now, since du is a perfect differential, we should have 

d*u _ _ B'ht 

BxBy~dyBx' 

B ( r -ds jv\ b 1 . r ds dv\ 



• 



X. 



FIRST RELATION 



BxBj 



Expanding 
dydx 



out and remembering that 



we get 



B*i> B z v 

dxdy dydx 



261 



and 



(4?: 



er ds _bt B± = &p<h_ bp b^ 

dx" By By "Bx 3x By By Bx' 

which when geometrically interpreted means that the corresponding 
elements of area whether represented on T, s or p, v coordinates are 
equal. This relation will hold for a simple homogeneous substance. 

For convenience of remembering we may write (47) as 
B{T*s) _B\p : b) 
d(x,y) 3{x,yj ' 

where '- '--stands for the determinant 
B{x t y) 



BT 


BT 


Bx 


dy 


Bs 


ds 


Bx 


By 



Any two of the four quantities T, s, p, v can be chosen as x t y. 
This can be done in six different ways and correspondingly, we have 
six thermodynainical relationships, though all of them are not 
independent. 

30. First Relation, — Let us take the temperature and the 
volume as independent variables, and put x = T, y = v in (47). 
Then 



BT _ , Bv_ _ . 
Bx By 



and 



$1 

By 



dv 



P = o, 

Bx 



(48) 



since T and v are independent. Hence we have 

(*) ={%) ■ ■ ■ ■ 

\ to/ r \3r } v 

which implies that the increase of entropy per unit increase of volume 
at constant temperature is equal to the increase of pressure per unit 
increase of temperature when the volume is kept constant. We can 
apply equation (48) to the equilibrium between the two states of the 
same substance. Multiplying both sides by T 



to It \dTf v 



(49) 



which means that latent heat of isothermal expansion is equal to the 
product of the absolute temperature and the rate of increase of 
pressure with temperature at constant volume. Thus, if a body 



262 



THERMODYNAMICS 



CHA1\ 



changes its state at T Q and absorbs the latent heat L, and the specific 
volume in the first and the second states are v x , v 2 , equation (49) 
yields 

L _ dp_ 
v*~-v. at 



or 



dp_ 
dT 



(50) 



This is Clapeyron's equation and is one of the most useful 
formulae in thermodynamics, A more rigorous proo£ of this will be 

given in sec. 39. 

Let. us first employ (50) to find the change in the freezing point 
of a substance by pressure. In the case of water at o C 1 

£, = 79.6 X 4,18 X 1A T ergs per gm. 
T = 273. 

1*2 = 1.000 e.e. (specific volume of w r ater at 0°C), 
0!= 1.091 c.c. ( „ „ „ ke at 0°< 

dp__ 79-6 x4-l 8x10- 
'* dT 27S x (1-1091)' 
Now if dp — 1 atmosphere = LOIS X 10 6 dyne/an 2 , 

8T= -0.0075°. 

This shows that the melting point of ice is lowered by increase 
of pressure, the lowering per atmosphere being 0.0075°. The pressure 

isary to lower die niching point by PC. is 1/.0075 = 133 
cm*. This accounts for the phenomenon of Regulation of ice 
and the experiment of Tyndall (p. 110). 

These results were quantitatively verified by Kelvin, His experi- 
mental results are given in Table L 

Table 1 .— Depression of freezing point with titer ease of pressure. 



Increase of pressure. 


Depression 


of 


freezing point in °C. 


observed. 




calculated. 


8.1 aim. 
16.8 „ 


0.0580 
0.1289 


0.0607 
0.1260 



For substances which contract on solidification the melting point 
will be increased by pressure. Thus for acetic acid whose melting point 
is 15.5°C de Visser found experimentally that an increase of pressure 
by 1 atmosphere increases the melting point bv 0,02432°C < , while 
equation (50) gave 0.02421 °C. This verifies relation (50), 









x.] 



FIRST RELATION 



263 



Equation (50) will also hold in the case of fusion of metals. With 
its help the change in melting point of any substance with pressure 
can be easily calculated. The values so calculated, together with the 
experimentally observed values, for a few metals are given in Table 2. 
The agreement is fairly satisfactory. 

Table 2. — Change in melting point with pressure. 



Metal. 



So 
Cd 
Pb 
Bi 



Melting 
point 

in °C. 



2 M 1 .0 
320.4 
326.7 
270.7 



Latent 
heat in 
cal./gm. 



in c.c. 
per gm. 



&T per 
1000 

atm. 
(cak.) 



1 t.25 
13.7 
5.37 
12.6 



0.003894' +3.34 

0.00564 | +5.9) 

0.0039761 +8.32 

-0.00342 -3.56 



ST per 

1000 
atm. 
(obs.) 

+3.28 
+6.29 
+8.03 
- &.55 



Equation (50) may also be employed to calculate the latent heat. 
For the vaporization of water at 100°C. we have the following data :-- 
T — 373°, ?'i = 1 c.c, w 2 = 1674 c.c. per gram. 

iL_ —27.12 mm. of mercury, 

a I 

. 373x(1674~l)x27-12*l-Q13xl0« ^ n ^ f 

" L ~ 760 x 4-185 xlO 7 '* 

The accurate experiments of Henning give L = 538.7 cal. per gin. 
at 100° C. which is very close to the calculated value. We can also 
calculate the latent heai, of evaporation of water at various tempera- 
turfs from (50) if we know the values of v 2 and ^ at these tempe- 
ratures. The values of L so calculated are given in Table 3, together 
with the observed values. The agreement is seen to be very close. 
Table 3. — Latent heat of steam at different temperatures. 



t°C. 



dp 



-&=. in mm. 



100 
110 
120 
130 
140 
150 
160 
170 
180 



dT 



of Hg, 



27.12 
36.10 
47.16 
60.60 
76.67 
95.66 
117.7 
143.4 
172.7 



v 2 in c.c. 



1674 

1211 
892.6 
669.0 
509.1 
393.1 
307.3 
243.0 
194.3 



L (calc.) 



539.5 
533.4 
526.6 

520.1 
512.9 
505.7 
497.5 
489.8 
481.8 



L (obs.) 
Henning 



03 



.7 
.1 
525.3 
518.2 
510.9 
503.8 
496.6 
489.4 
482.2 



"Taken from Jellinek, Lehrbuch dcr Fhysikalischcn Chetnk, Vol. 2, p. 517 
(1930 edition). 



264 



THERMOD YNA MICS 



[chap. 






31. Application to a liquid film, — Equation (48) may also be 
applied to the case of a liquid film. If such a film is. stretched, its 

volume remaining constant, the energy equation for Lhis case is 

($Q) s = du-2<rdA, 
for the work done by the film is -2odA, where <x is the surface tension 
and dA the increase in area of the Film, The usual term pdv on the 
right has disappeared since the total volume remains constant. Hence 
corresponding to p and dv we have in this case -2,q and dA respec- 
tively. Therefore equation (49) yields 

\m)t,v = " 2T \dT)»,A 

and for a finite change 

•q—it (£),*• (si) 

For a liquid the surface tension decreases with temperature, 

therefore — r is negative and 8£> is positive. Hence an amount of 

heat must be supplied to the film when it is stretched in order that 
its temperature may remain constant. In an adiabatic stretching the 

temperature will fall by an amount 

«■- ££(£■.) dA (52) 



where C A is the heat capacity of the film. 

32. Second Relation, — Another important application of the 
thermodynamic formulae consists in their application to some adia- 
batic changes such as the sudden compression of a liquid or sudden 
stretching of a rod. For this case let. us put x = T, y = p. Equation 
(47) then reduces to 

&)r — (£).• • • • ™ 

which means that the decrease of entropy per unit increase of 
pressure during- an isothermal transformation is equal to the increase 
of volume per unit increase of temperature during an isobaric process. 
Multiplying both sides by T we have 

{£),=- r (ia=- r - ■ • < m > 

where a is the coefficient of volume expansion at constant pressure. 
From tins relation it follows that if a is positive, i.e., the substance 

expands on heating, | jM is negative, and hence in this case an 

amount of heat must be Laken away from the substance when the 



*■] 



SECOND RELATION 



265 



pressure is increased, in order that the temperature may remain 
constant. That is, heat is generated when a substance which expands 
on heating is compressed. For substances which contract on heating,, 
a cooling should take place. 

These conclusions were verified experimentally by Joule who 
worked with fish-oil and water. The liquid was contained in a vessel 
closed at the top by a piston and pressure was suddenly increased 
by placing weights upon the piston. The change in temperature was 
measured by a thermopile. From (54) we can deduce the increase 
in temperature AT produced by a sudden increase of pressure A/>- 
We have* 






PB» 



where c$ is the specific heat of the substance and v its specific volume. 
We have assumed that v a /c p is independent of pressure, and then 
integrated for a iinite change /\,p in p. Joule's results with water are 
very interesting and are given in Table 4. 

Table 4. — Increase i?j temperature of water by sudden 
increase of pressure* 



Ap in kg. per cm 2 


Initial temp. 
in °C, 


AT (obs.) 


AT (calc) 


26.19 




- 0.0083 


- 0.0071 


26 J 9 


5.00 


+0.0044 


+0.0027 


26,19 


11.69 


0.0205 


0.0197 


26.19 


18.38 


0.0314 


0.0340 


26,19 


30,00 


0.0541 


0.0563 


26.17 


31.17 


0.0394 


0.0353 


26.17 


40.40 


0.0450 


0.04 7 U 



The agreement between the observed and the calculated values 
is seen to be very close. This proves the essential correctness of the 
theory ; in fact, these results formed one of die earliest experimental 
verifications of the second Jaw. Thus the thermodynamic theory 
explains the remarkable fact that water below 4°C. cools by adiabatic 
compression inspite of the fact that the internal energy is increased. 

Another series of experiments consists in the adiabatic stretching 
of wires. The best results were obtained by Haga. The change in 
temperature of the stretched wire was measured by means of a 
thermopile formed by the wire itself and another thin wire wound 
round it. It will be seen that tension means a negative pressure 
and hence wires of substances which expand on heating should show 
a cooling when stretched adiabatically. In this case the work done 

*We may directly deduce this result from equation 



266 



'III liRMOD YN AM ICS 



dv 



iCtJAP. 



is not pdv but -JPctt hence ?=. must be replaced by ^ where I is 
the length of the wire. Hence in place of (55) we get 

where /3 is the coefficient of linear expansion, ta the mass per unit 
length of the wire, c p its specific heat in mechanical units and C the 
heat capacity of the wire. 

For a German silver wire of diameter 0.105 cm. at room tempera- 
ture, Haga found the mean value AT = -0,1063 for a tension of 
13.05 kg. and AT = -0.1725 for 21.13 kg. tension, giving a mean 
value of 0.00813 per kg. If we substitute the values o£ w, c p , /3 in 
equation (56) and use the value / — 4.1S X 10 7 ergs per tab, we 
get AT =-0.00810 in dose agreement with the experimental value. 

India-rubber when moderately stretched has a negative expan- 
sion coefficient. This should show a heating effect when further 
stretched adiabatieally, which is found to be true experimentally and 
may even be felt by the lips. 

33, Other Relations.— Besides the above two, there are other 
rtant relations. Thus putting x — S* y = v in (47) we act 

IS).-- (ft- • • ■ ^ 

Again putting X ss s, y =/j we get 



This is the third 



i 9 —) = (— ) 



(58) 



This is the fourth relation. The interpretation of these results is 
left to the reader. 

The above relations are known as Maxwell's four thermodynamic 
relations. Besides these, there are two more relations which may 
be obtained by taking p, v t Or T, j, as the pair of independent 
variables. They are 

GH.Or-eirew.- 1 - • ■ « 

These arc called the fifth and sixth relations and are much less used. 
34. Variation of Intrinsic Energy with Volume.— From the tela- 



tta(£) r -(§L) p . we find by substituting ^±^' for 



ch 



x.] 
that 



VARIATION OF INTRINSIC ENERGY WITH VOLUME 



©rT*{fr).-* 



267 



(61) 



This equation enables us to calculate the variation of intrinsic energy 
with volume. 

For perfect, gases* p = y, hence |^J ^ = (Joule's law, p. 47). 



BT 



For gases obeying the van der Waals' law p = ,^» — y\ it can 
be easily seen by substituting in (61) that 

For general systems, we can transform (61) to a form which allows 



du 



us to calculate J- from experimental results. This follows from 
do 



(68) 



equation (IB) of page 240 where it is shown that 
\dTt,r \BvlT\dTlp 

= TEa-Pi ..... (64) 

where £ = bu1k modulus of elasticity, « = coefficient of volume 
expansion. 

We shall compare the ratio -. u : p for a typical liquid, viz., 

mercury. For mercury at 0°C and atmospheric pressure we have 
as on p. 240. 

(dP) - 46.5 atm. em-* ^C- 1 
1(|"\ = 46.5 X 273.2 - 1 = 12700. 



Hence 



du 



Now — is sometimes known as the internal pressure the idea 

being that the visible pressure is equal to the force per sq. cm, 
with which the particles bombard a layer inside the liquid minus the 
internal pressure, i.e. force/cm 2 with which they are drawn inwards 
due to forces of cohesion. The above comparison shows that for 

* U, V refer to a gram-molecule, hence ay = ~qZ m £ e » el 'al. hi the text 
both forms have been used. 



TH ERMODYN AMICS 



CHAP. 



ordinary liquids, this force is very great but when we increase the 
temperature, the ratio diminishes till it vanishes when the perfect 
gas stage is reached. 

35* The Joule-Thomson Effect. — In Chapter VI, we proved that 
in the Joule-Thomson expansion u -J- pv remains constant while the 
pressure changes on the two sides of the throttle valve. Let us 
calculate the change in temperature due to an infinitesimal change 
in pressure during this process. We have by the conditions of the 
process 

d (u -f pv) — (65) 

Since from the two laws of thermodynamics du -\- pdv ss Tds, we 
have from (65) 

Tds + vdp = Q, .... 



(66) 



or 



\ QW 



Hence 



or 



r (a 8 V)/ T+T (|) r * +t * = °- 

T \dT) p ==cp; [dpfr^ " XdTlp 



rlBv \ 



\dp )h c„ 



— o 



(67) 



- — | . 

op Ik 
For a perfect gas, TK J - V is easily seen to be zero and the 



Joule-Thomson effect vanishes. The porous plug experiment therefore 

provides a very decisive test of finding whether a gas is perfect or not. 

For a gas obeying van der Waals* law it can be proved that 

1B_T\ = 2a{V-b)*-bV*RT V 
[dp 



(B_T\ = 2t 

W hi = 1 



RTV*-Za{V— bf C; 

-idr-*) 1 **" 



JVOX, 



(68) 



ST 



Hence ^— is positive, Le. when &p is negative there is cooling 

as long as 2*<t„- . At T greater than ZafbR the gas becomes 

heated on suffering Joule-Thomson expansion. The temperature 
stands for the total heat u + pv or the enthalpy of the system. 



x.] 



CORRECTION OF GAS THERMOMETER 



269 



given by T. — 2a/bR is called the temperature of inversion since on 
passing this temperature the Joule-Thomson effect changes its sign. 

It will be easily seen that —=—T s approximately where T e is the 

critical temperature of the gas. We have already dwelt upon this 
point in Chap. IV, These results have been approximately verified 
by experiments, 

36. We will analyse the Joule-Thomson effect in another • 
which perhaps gives a better insight into the mechanism of the 
phenomena. We have 



Now 



Tds = du 4- pdv, 

. , l*Z\ - _/*l\ ( S (P»)\ 

" M3> /*" Wt \ dp IT 



(69) 



The first term on the right, measures deviation from Joule's law 
while the second gives the deviation from Boyle's law. and the 
Joule-Thomson effect is the resultant effect of deviations from 
both these laws. 

Now {-'-) is always negative, for that part of the internal 

energy which is due to molecular attractions always decreases with 
decrease of volume, Le., increase of pressure. Hence due to devia- 
tions from Joule's law alone the Joule-Thomson effect will be a cooling 
effect. Upon this will be superposed the effect of deviations 



from Boyle's law which is a cooling effect, if 



d(pv) 
dp 



is negative (t.«r v 
&f» 



before the bend in Fig, 6, p. 98) and a heating effect if —fr' 

is positive. 

37. Correction of Gas Thermometer, — Equation (67) can be 
directly utilised for giving the absolute thermodynamic scale from 
observations on an ordinary gas thermometer. For we have 



'fr).-*GV 



*P 

where all these quantities ought to be measured on the thermo- 
dynamic scale. In actual practice we use a gas or any other thermo- 
meter to measure this temperature which we may denote by 6 
and c/ denotes the specific heat on this arbitrary scale. We have 

_ dO _ dCL d9_ , </0 

6 *~~ df d& dT cp dT' 






270 



Further 



TIIERMODYN AM ICS 

ldT\ _dT tm 

\dplk d$ \dpi h 
tar/," \dBfpdT' 



[CHAP. 



Hence (67) yields 



dT 
T 



(!) 






v+c. 



\Bpn 



or 



(70) 



The quantities occurring on Lhe right-hand side can be measured on 
any thermometer ; all that is necessary is that the same thermometer 
should be used in all these measurements. Further these quantities 
vary with temperature. As a first approximation* let us assume 
them to be constant. Then on integration we get 

log T = log (v + pc/) + const. 

dp h 

constant. This shows that the thermodynamic temperature T is 
related to the volume of the gas in this complicated way and not 
only as T qc v as in the case of perfect gases. Let the thermodynamic 
temperatures corresponding to melting point of ice and boiling point 
and Tioo respectively and the corresponding volumes 
of the gas he u and v tm . Then 



,, stands fior the Joule-Thomson effect r«— ) and 



a is a 



T"n 



^ 1IM ' 



100 



'j ; :'-'"; 






be the volume coefficient of expansion. Then 



r.-- L(H-fi«l). 



(71) 



1 Ins gives the temperature of the melting point, of ice on the thermo- 
dynamic scale. 

The results for a few gases have been calculated and are given 
in Table 5, For hydrogen the Joule-Thomson cooling is -0.039°C. 
per atm., Me t = 6.86 X 4.18 X 10 7 ergs per mole per °C, Mv = 
22.4 X 108 c, c , per mole and a == .003661 S per D C. 



* More accurately, however, we have to integrate over short intervals. 



*■] 



.'. T = 273-13 ( I — ' 



EXAMPLES 



039x6*86x4-18x10 



271 



! ) 



10 e x 22-4x1 
275.13 (1 -.00050) =273,0 degrees. 

Table 5 Joule-Thomson correction to the gas thermometer. 









Uncorrected 








Volume 


Mean Joule- 


temperature 


Correction 


Thermo- 


Gas. 


coefficient 


Thomson 


of melting ice 


term 


dynamic 




of expan- 


cooling per 


fl 1 


e p ' se 


tempera- 




sion a 


atm. 


#o = - 


0% dp 


ture T 




per °C. 










H-, 


.0036613 


- 0.039°C. 


273 J 5 


-0.13 


273.0 




.0036706 


+0.208 


272.44 


+0.70 


273,14 


co 2 


.0037100 


+1.005 


269.5 


+4.44 


273.9 



For air the dnin are most reliable and yield the value To = 273.14°. 
It will be seen that though for the various gases the melting- point 
on the uncorrected gas thermometer (0 O = l/« in column 4) is much 
different, the melting point corrected by making use of the Joule- 
Thomson effect comes to about the same value, viz., 273°, 

To find the correction to be applied to the gas thermometer at 
other temperatures we must employ (70). Thus comparing ^ (70) and 
(71), we have a .= l/v a. Then any temperature T t is easily found 
if v., measure the corresponding specific volume and the Joule- 
Thomson effect ; for 



r,-+(* 



+f 



,de\ 



(72) 



The eas thermometer temperature is 9i s= — - — , and the correction 
° v a a 

. ' -aa 

term is — *- - r . Rut as already mentioned on paste 10 the existing 

v a a dp l * 

data on Joule-Thomson effect are not sufficient, and the corrections 
are usually calculated from deviation from Boyle's law. 

38. Examples. — We will now give a few examples which will 
illustrate the utility of the foregoing thermodynamic formulae and 
will give some practice in applying them, 

1. Prove that c p - c v = ^(^) (|y) = TE#v, where E h the 

bulk modulus of elasticity, a the coefficient of volume expansion and 
v the specific: volume. 



272 



THERMODYNAMICS 



CHAP, 



*■] 



:- 'I I CIFIC HEAT OF SATURATED VAPOUR 



273 



["»-+-rfr) r Tb) m+ T%) r l») t 

r feh— fc)r-(fr). 

••• "— (Ir).(fH 

[ We have |^J ^j ^) , Differentiating partially with 
respect to T we get ^ = ^ , Now * p = 7" ( | J f ) and therefore 

U» It ' Bv5T §Tdv ~ l tar*/ J 



3. Prove that 



Stan From 



(bp)t = ' (bt) p and P rocecd * above - ] 



39. Clapeyron's deduction of the Claasius-CIapeyron Relation.— 

equation was deduced on p. 262 but the method emploved there 
is open to objection because Lhe thermodynamic relations hold rigor- 
ously for a homogeneous substance and 'their extrapolation to dis- 
continuous changes is open to question. As however the relation is 
important, a more rigorous deduction is given below which is due to 
Ciapeyroii. 

Let ABCD, EFGH (Fig. 4) represent 
two consecutive isotherm ah at tempera- 
tures T and T + dT. From F and G 
draw adiabatks meeting the second 
isothermal at M and N. We can 
suppose a unit mass of the substance 
to be taken through the reversible Carnot 
cycle FGNM, for Instance, allowing it to 
expand iso thermally along FG, adiaba tic- 
ally along GN and compressing it along 
NM isothennally and then adiabatically 
along MF. The substance at F is in 
the liquid state and at G in the form 




Fiff. 4,- Clapeyron's deduction 

ui tlit Clausius-Qapeyron 
relation. 



of vapour. The amount of heat taken during the cycle is therefore 
L 4- dL at temperature T + dT. Therefose the work done during the 
cycle is from equation (10), p, 216, given by 

f T . m/ T+dl-T\ r , _ 




■ 

: uiswiixce ueiween e\r anu jvii\, t,e. f tip tne increase u i pressure 

to increase in temperature by dT, change in 

volume due to evaporation or I grain of liquid and therefore equal to 
ya-Vj where v % , tr a denote the specific volumes of the vapour and 
;>!: liquid respectively. Hence the area of the cycle is 

Equating the two expressions for area we get 

{V % -V t )dp~ i^dT, 



or 



(50) 



_ L 

Equation (50) h called the First Latent Heat equation or the 
late; ion of Clapeyron. 

40. Specific Heat of Saturated Vapour.— -A simple expression for 
the specific heal of saturated vapour may be. deduced with the help 
of Fig, 3. Consider the cycle represented by the curve BFGC. The 
amount of heat taken by the substance during its passage from Pa to 
F is c tj dT and that during the pal s L -{- dL. On the other 

hand Che substance gives back heat equal to c dT during the path 
GC and equal to L during the path CB. c t denotes the specific heat 
of the liquid in contact with vapour, and e r the s; of the 

vapour in contact with liquid (specific heat of saturated vapour). 
The total amount of heaE taken during th c; BFG i 
c h dT 4- L -f dL - c s dT - /., 

and this must, in the limit be equal to the area FGNM, which is 

UT 



equal to 



T 



as proved in the last sei 



■ T+L. 



^- % JT-L-- 



or 



= c, ~e. 



(73) 



L 

dT ~ r 

This is called the latent heat equation of Clausius or the Second 
Latent Heat, equation. 

18 









HMOUYNAMlCS 






It may be noted that c s% is neither the specific heat at constant 
p. inc heat at constant volume. Here the liquid 
and die vapc rays remain in contact and therefore the vapour 

always remains saturated, Both the pressure and the volume : 

| that the condition of saturation is always satisfied, It is easily 
seen that c s does not appreciably differ from c^ the specific heat 
at co pressure, for the effect of pressure is too small bo brine 

about any considerable change in the state of the liquid. c tj ea 
therefore be put equal to c H . We can now calculate the value of i 
from equation (73) . For water at 100°C. 

= -0-64 cal. gmr^C" 1 ; L = 539 cal. gm." a 
T =-- 373* : c. = 1-01 cal gmr^G" 1 



can 



c t = 101 

s 



0.64-— = -1*07 caLgm^C-' 



Thus the specific heat of saturated water vapour at !00 fl C. comes 
out to be a negative quantity. This is rather radoxical Tesult 

ame lime perfectly true. In Chapter II we have seen 
ic heat may vary from + * to - oo depending entirely 
In the present case the conditio 
satisfied. 
i, at 10O°C, and 787.5 mm. at 101 *C. 
the sp ' ater vapour 

sated to 101°C. at constant pressure it 
■ the condition of saturation has to be 

be compressed till the pressure becomes 787.6 
rhis compression generates heat, and in the case of water at 100*C. 
the heat generated is so great that some of it must he withdrawn in 
order that the temperature may not rise 101*. The net result 

in this case is that heat must be withdrawn from and not added to 
the system during the whole operation. Thus we expla 
why the specific heat of saturated vapour sometimes becomes negative. 
' The same idea can be expressed mathematically. We have 

refers to the condition oE saturation being satisfied. 



Or 






e t = e. 



T 



r , v MM 
\ST) t \dTl m . 



x. 



THE TRIPLK PC 



275 



Now for all vapours 






is positive and hence c, is always less 






than Cp and may even become negative. 

From these considerations it will be seen that saturated steam 
must become superheated by adiabatic compression, e.g,, water vapour 
at 100°C. and 760 mm. pressure., when compressed suddenly to 
787 mm. would be heated by 2. PC, and hence become superheated; 
in other words, when the temperature of saturated steam is raised 
it gives out heat. Conversely, when it is allowed to expand adiaba- 
tically, say, from 787 mm. to 760 mm., the temperature would fall 
by 2.1 "C.j Le., to 98.9°, and hence it would be supercooled, and 
partial condensation may occur. For certain vapours such as saturated 
ether vapour, the work done in compressing the substance is not so 
great and the specific heat is positive. These do not become super- 
heated by adiabatic compression. 

These conclusions were experimentally verified by Him in 1862. 

He allowed steam from boilers at a pressure of five atmospheres 
(temperature about 152°C) to enter a long copper cylinder fitted witii 
glass plates at its ends. When all the - air and ' condensed water 
had been driven out, and the cylinder had attained the temperature 
of the steam, the taps in the supply and the exit tubes were clos 
and the vapour when viewed from the ends looked quite transparent. 
The egii tube -was next suddenly opened and the vapour expanded 
adiabatically and a dense cloud was observed inside the cylinder. The 
cloud however soon disappears as the cooled vapour rapidly absorbs 
heat from the walls of the cylinder which are at 152°C. No such 
condensation was observed in experiments with ether vapour. 

41. The Triple Point— If we now plot the saturation value of p 
against T we get a curve OA (Fig. 5) the slope at every point of 
which will be given by 

£, 

dT T{v s -v t ) » 

where L t is the latent heat of vaporisation, and v g} v) denote the 
specific volumes at the temperature considered. 

To fix our ideas let us consider the case 
of water. When the temperature is re- 
duced to 0°C. water freezes and we get 
ice. But ice has also a definite vapour pres- 
sure which has been measured. The vapour 
pressure curve of ice may be represented 
by the line OB., the slope at every point 
of which will be given by 

dp L, 

dT T(v t -v,) 

where L s is the latent heat of sublimation. 




Temperature 
Fig. &— The triple point 



276 



THLRMODYNAMICS 



[CHAI\ 



x-1 



ii ie TKirLi': PO] 



277 



Similarly lor the phenomenon of melting, the curve OC repre- 
sents the relation between pressure and temperature, and the slope at 
any point is given by 

\ dT T(v,-v t ) ' 
where L s is the latent heat of fusion. We have already seen on 

p, 262 that for ice -?• — 133 X 10° dynes cm. - °C-\ i.e., the curve 
1 a r 

should be almost, vertical. 

For substances which contract on solidification the slope of OC 

will be positive. The three lines OA, OB, OC are respectively called 
vaporization line, sublimation tin •, and in the parti- 

cular case of water they are called steam line* hoar-frost line and ice 
line respectively. 

Consider the substance in the state represented by any point Q 
above the line OA. It will be noticed that tile pressure of the subs- 
tance In this state is greater than that which will correspond to the 
saturated vapour pressure at that temperature and which is given 
by the intersection of OA with the ordinal a from Q. At 

this iressure represented by O, therefore, die substance can 

not boil at this temperature as boiling point is raised by pressure, 
and mi' lore exist as liquid. Thus the region above OA xe- 

pres, Similarly for points below OA the correspond- 

too low and the substance must exist as gas. 
■ substance r • as solid and below it 

any point in OC due to the pressure being Larg 
g to the ice line, ice will melt and therefore 
presents water, and that below represents ice. 

It can be easily shown that these three curves must meet in 
a single point which is called the triple point. For, if the curves do 

not meet at a point, let them 
intersect each other forming a 
small triangle ARC (Fig. 6) - 
Then since the space ABC is 
above AB it must represent the 
liquid region, but it. is below 
AC and must therefore repre- 
sent gas, and is also below 
BC and must therefore also 
represent solid. Thiis the re- 
gion ARC must simultaneously 
represent the solid, liquid and gaseous states. Evidently it is 
impossible to satisfy these mutually contradictory conditions and the 
only conclusion to be drawn is that such a region doesnot exist. In 
other words, all the three curves meet in a single point. 

The coordinates of the triple point can be easily calculated from 
the consideration that at this point, the vapour pressure of water 




jtjjt. 6.— Impossible intersections of 
ice, steam and hoarfrost lines. 









is equal to the melting pressure of ice. The vapour pressure of 
water at 0°C is 4.58 mm. and at 1 C G, 4.92 mm. Thus the vapour 
ure rises by 0.34 mm. per degree, and therefore, if I is the 
triple point the vapour pressure p at the triple point is given by 

p = 4.58 -f 0.34 t (74) 

The melting pressure of ice at 0°C is 760 mm. and the change of 
melting point with pressure is 0.0075°C per atm. Therefore the 

melting point t at the pres.su re p will be 



t =t 0.O075 - 



0.0075 

~76CT 



P- 



(75) 



Solving (74) and (75) we get 

t = 0.007455 C C and £ = 4.5824 mm, 

It will be apparent that since the change in vapour pressure is too 
small in comparison to the change in melting pressure, it is useless 
to go through the complicated calculation given above and we can 
simply assume that p = 4.58 mm. approximately (as t will be very 
near 0°C) and calculate the corresponding melting- temperature. Thus 

( = 0.0075- :0.007455»C. 

The coordinates of the triple point are therefore t — .0075°C, p 

mm. At this point three phases (solid, liquid and vapour) co-exist. 

It was former J y supposed that the curves OA, OB are continuous. 
first proved by Kirch holt that this is not so. for according to (50) 

OA at 0°G =■ 



£ far 



dT ' ~ T{v t -v^ 

607 x4 1 8 i :• .760 

— 273x21 xl0*xl0" • 

= 0.337 ram. per degree. 

dp , ^ , «i • \ 687x4.18x10?;: 
^ for OB (subhmatipn) = ~ m ^ ^ ^ w 

— 0.376 mm. per degree. 

The dotted curve OA' is merely the continuation of OA, It 
represents the vapour pressure of supercooled liquid. At - 1°C\ we 
have vapour pressure of liquid - vapour pressure of .solid = .04 mm. of 
mercury. This has been verified by the expe tits of Hoi born, 

Scheel and Henning. 

Boohs Recommended. 

1. Fermi., Thermodynamics. 

2. Planck, Treatise on Thermndynam : - 
H. Hoaie, Thermodynamics. 

4. Smith, The Physical Principles of Thermodynamics (1952), 
Chapman & Hall. 

5. Epstein, Text-Book of Thermodynamics, John Wiley. 



CHAPTER XI 



RADIATION 

1. Introduction. — Even when a heated body is placed in vacuum, 
it loses heat In tin is case no heat can lie lost by conduction or 

convection since matter, which is absolute bote 

processes, is absent. In such cases v that heat is lost by 

'Radiation.' To differentiate this process from conduction, it is 
enough to note that copper and v lich are ho much different 

in their conducting powers, cut off radiation equally well when plated 
between the hot body and die observer. 

Now heat has been shown to be a form of energy, and die pro- 
pagation of heat by radiation consists merely in a transference of 
energy. But the radiant energy in the processes of transference does 
not make itself evident unless it falls on matter. When it falls on 
matter and is absorbed, it is converted to heat and can be thereby 
detected. 

Let us now study some of the p les of radiant For 

this purpose we require a soxin it as radiator and some h 

ment to measure the emitted radiation. 
As emitter Leslie employed a hollow i 
cube filled with hot water, whose sides 
coul- i with different substances. 

The cul" tnade that it can be rotated 

about a vertical axis. Such a cube 
called a] uhe and i in Fig. 1, 

For die radiation 

expi nployed a differ* 

thermometer or a thermopile which are 
described in the next section. 

2. Some Simple Instruments for Measuring 
Radiation. — Leslie, one of the earliest workers, 

•■'ir-thermmneter which is 
now only of historical interest. This consists 
of two equal bulbs A and B (Fig. 2) 
ing air and communicating with each other 
through a narrow tube bent twice at right a 
and containing some non-volatile liquid like 
uric acid. When the bulbs are at the 
same temperature the liquid stands at the same 

at in both the columns C and D, but if one 
of the bulbs level of the liquid 

i in the column C and rises in the column D, 
By noting- the difference in level of the two 
columns the difference in temperature of the 
bulbs can he easily calculated. 

Melloni was the first to introduce thermo- 
piles for the measurement of radiant energy. In its original form. 





Fir. 2.— Leslie dif 
fitlal air tlicr- 
rn.:inn ter, 



XI.] 



PROPERTIES OF RMUANT ENERGY 



279 




3,— principle of the 
ale, 




•ofa thei consisted of a number of bismuth and antimony 

Is jmned> as indicated m Fig. 3. 

The left face forms the hot junction, and 
the right the cold junction. The antii 
bars and the bismuth bars 

ihe current flot 
iio t juncti o n from bismu th to an 

u rmopile a 1" 

i : are 

arranged in the form of a cube such that 
he hot junctions are at one face and the 
cold junctions at the opposite face. Such 
a cube is shown in Fig. 4. The near lace 

iction and is cos ck. 'lire thick aote the 

;„,,; tti attl, generally mica, 

the differeiii layers, and the thin lines 
denote the : junctions. The current 

across the .soldered junc- 
itimony bar, 
liters the next layer at B on 

. Finally the cur, 
m opi] thermoj 

inted for use is shown in Fig. 5. 

is provided with a com 
rotect from stray radiatio 
n..r] ,i .. : | fitallic 

I the hot junction whi in use, 

Other ' 

cribed in sections 21-24. 

3. Properties and Nature of Radiant 
Energy.— With the aid bl these sii 
apparatus, it can be demonstrated that 
Radia and Li#i ' 

, and h;' ■ >perticB. We 

mention he points of res 

blance. 

(i) Radium energy, 

cuum, for we arc 
i a hot eh < . : 

lam] ' fi - is highly 

: I I current passe; through 

the fi lament which is not -'- to 

li glow, the thermal 

radiation coming from tl r can be 

dete nsiti+e thermopile. 

lion, like Is in 

straight lines, — This can be easily 
for the heat coming from a flame can be easil) ci oi mterpoi 



Fig-. • 




$r— The thermopile 
(mounted). 






280 



RADIATTO-V 



[CHAF. 



_.. 





i-U 



Fig, tf.— Verification of : 



a screen just sufficient to prevent the light coining from ii. On replac- 
lie flame by a hot noi :t, no heating effect can be 

detected by a thermopile. The geometrical shape oJ 

can be verified by cutting out a cross from the screen and holding 
close to it a piece of wood coated with paraffin wax. The wax will 
melt in the shape of a cross. 

(Hi) Radiation travels with the velocity of light* This follows 
from the observation that the obstruction of the radiation from the 
sun during a total solar eclipse is immediately accompanied by a fall 
of temperature. 

(iv) Radiant energy follows the law oj inverse squat; ,,, tight, 
— This may be experimentally proved by the simple arrangement 

shown iu Fig, 6, A is a vessel 
containing hot water and having 
one surface B plane and coated 
with lamp-black, S is a 
pile. Tt will be found that if we 
move the thermopile to another 
position S' (say double Ehe pre 
ius distance) , the del lection of 
the galvanomeb :d to 

it will be unchanged 
area of die surface of B from 
which radiation can reach the 
d four time amount of radiation 

intensity of radiation un- 
reduced' to ii; 

can be easily 
g two 
parabolic mirror 
and B at some disl 
once apart with their 
axes in the same line 
(Fig. 7) . A luminous 
source placed at the 
i produces 
an image at the ! 
of B. If we replace 
the luminous source 
by a hot non-luminous 
one, say a Leslie cube, 
and put a piece of tinder at the other focus, the tinder is ignited. 
This shows that heat and light travel along the same path and hence 
obey identical, laws. In fact all the laws ""and results of geometrical 
• to it, 




Fig. 7. — Verification of laws of reflection for 
radia 






XI. 



OF RADIANT ENERGY AND LIGHT 



281 



■') Radiant enerp rum 

may be obtained by retraction through a prism (sec. 29), 

(-(■ii) Thermal radiation exhibits the phenomena of interference 
and d: ■■■'. — Diffraction can be easily observed with the he 

a concave grating or by employing a ti ission grating made of a 

number of equidistant parallel wi 

(vii radiation can also be polarised in the same way 

as light by transmission through a tourmaline plate. This can be 
. by the following experiment : — Arrange that Kght from an in- 
candescent lamp traverses two similar plates or tourmaline. On 
rotating one of the plates the image will be found to vanish. Replace 
the lamp by a Leslie cube. Now a thermopile placed in the position 
occupied by the image will record no deflection ; if however one of 
the plates be rotated a deflection will be observed. 

4. Identify of Radiant Energy and Light — Continuity of Spectrum. — 

All these observations lead us to the conclusion that radiant energy 
is identical with light. As we shall see presently, Radiation or 
Radiant Energy is the general and more expressive term, 
rather visible light.) being only a kind of radiant energy which has the 
distinctive power of affecting the retina of the human eye, and thus 
producing the sensation of colour. But like other kinds of radiant 
energy light is also converted to heat when it is absorbed by -mi 

The identity of light and radiant energy can further be seen from 
the ft >1 ' owi 1 1 g v. Kpe rimen ts : — ■ 

When we prod- 1 ctrum of the sun by means of a prism T 

it ts terminated on one side by the red, and on the other side by 
the violet. But it can be easily seen that these limits are only 
apparent, and are due to the fact that the human eye is a very 
imperfect instrument for the detection of radiation, W. Herschel* 
n'lSOO) placed a blackened thermometer bulb in the invisible 
beyond the red. and found that the thermometer recorded a rise in 
temperature. The rise in temperature was observed also when the 

mometer was placed in the visible legion. Thus W. Herschel 
discovered the infra-red part of the solar spectrum, and showed that 
it was continuous, with the visible spectrum. 

The source of radiation in this case is the sun, which may be 
regarded as an intensely hot body. But it may be substituted hj 
••uffidently hot substance, say a piece of burning coal ; the posi- 
tive crater of the arc, or a glowing platinum wire, Only in th ■ 

the intensity of radiation is not great, mid the spectrum does 
not extend so far towards the violet. 

Hi) n of light by -.-~A very simple experiment 

suffice to bring out the point which has been just mentioned. 

Suppose we take a piece of blackened platinum wire and pass through 

n Herachel (1738—1822) was bom at Hanover but settled in England. 
I worked as a musician and later became ski astronomer. He discovered 

• planet Uranus in 1781 and later discovered! the infra-red radiations. 



282 



RADIATION 



[CHAP. 



XI. 



EI.ECTROMAGNETIC WA 






it a continuously increasing current. The function of the current 
is simply to heat the wire. We find that it becomes warm, and 
sends out radiant energy, If a thermopile be held near it, t*M 
nometer connected to it will show a deflection* When a slightly 
stronger current is passed the wire begins to glow with a dull red 
light. This shows that the wire is just emitting red radiation o£ 
sufficient intensity to affect the human eye. Accurate observations! 
show that this takes place at about 525 ,J C (Draper point) . With 
asing temperature, the colour of the emitted radiation changes 
from dull red to cherry red (900 D C), to orange red (1100°C), to 
yellow (125G*C), until at about I600°C it * es white. Thus the 
temperature of a luminous body can be roughly estimated visually 
from its colour. L Such a colour scale of temperature is given in the 
chart on pp. 25-26, This shows that si waves are emitted by a 

d body in sufficient intensity only with ini 
Vice versa, we may argue that when the temperature of the wire 
is below the Draper point, it is emitting longer waves than the 
red, but these waves can be detected only by their heating effect. 

Radiant Energy or Radiation is thus a more general name for 

Light. It can he of any wave-length from to co as illustrated 

ie chart i s 283-284. light forms less than 8/4 ths of an 

■ " 
Ex m of the Chart. — We here the mic 

1 along the vertical line. Thus (-5) on 
he wavelength is 10- B cm,, (—2) inda 

hart visible light extends from 

;l4x lHon. (violet). The infr; 

W. Hei .tension towards the 

tb side is chiefly due to die reststrahlen method of Rubens 

and a are detected chiefly b thermopile, 

for or plates are not sensitive beyond the green. 

romatic plates may be used, but they are also 

i p. mi beyond 8000 A. U. Beyond this plates dipped in 

or neocyanine may be used up to 10000 A. 17. Recently 

photographic plates sensitive up to 20000 A. U. have been placed on 
the market. 

The Hertzian waves are produced fay purely electromagnetic 
ids (due to oscillation of current in an indue ►acity cir- 

cuit) and were first discovered by Hertz in 1887. Marconi ap 
i for wireless transmission (1894). Waves used for this put 
are generally 10 — 500 metres in length, though waves of about. 10 
length have been used in the past. These long electromagnetic 
from the light waves only in their wavelength, and 
i :4s have been made to get shorter and shorter waves by purely 
electromagnetic methods so that one can pass continuously to the 

f According to a recent experiment by Bs . persons who have r 

their eyes in the dark carefully, can detect radiation even from a h 
to about 400 °C. 





ELECTROMAGNETIC WAVES 


Wavelength 


Log 


Radiation 


Generation 


Detection anc 
Analysis 


Discoverer 


cms 












Siegbahn 
unit 


-11 


Hard 
Y rays 


Reactions 


Paii- | 
production 

effect 






-10 ^ 










•0GA.U- 


'". 


y-rays 

> 


Radioactive 
Disintegration 


Photo -electric 

■:,hcd 




10-*' 
Angstrom 
unit 


] X-rays. 
Hard 

~ 8 ' Medium 


Bomhardment 

targets by 
electrons 


Cry* 
Photography 

charal 


jttiintgenue&J 
Laue mm 






-7 


/ Soft 




i 
i 


Compton[i9£5! 
ThitmudiUtefl 




136 A,U. • " 6 *n 
1-0XKP-5 •jL t ^ k 


Vacuum Spark 

Spark. Arc 
KLame 


Photogr 
aphy . 

Pri: ma 

. a 

: •:: ■.-- 

._: 9 - 


1 

IS 

~ .'- 

11 

8 


Millikan ctsi9> 

& 
Lyman ami 

Schumann 

■::■:. J: 

RitterdMfij 

NEWTON 

HerSchelO&iO) 






-3 


1 Infra-red 


Heating 


Wire gratings 

& Tharroo- 

cooples 






1 mm - 


-2' 






Residual ray 

Method 


Rubens & 

Nicholsassa) 




'4 mm ■ 
4mm- 


J 
-1 

0. 


J Hertzian 
J waves 

1 SI Dl 


Spark gap 


Coherer 


Wood ana 

Nichols & 
Tears (wao 

BoBeoffla J 





















mm 



284 



RADIATION 



CHAP. 






XI.] 



FUNDAMENTAL RADIATION l'UG< 






ELECTROMAGNETIC WAVES *»w 


Wavelength 


b. 


Radiation 


Generation 


Detection and 
Analysis 


Discoveror 


cm 


s-r 

















Spark gap 
discharge 








+i 


Hertzian 
waves 


Triode valve 


Radar 

(Micro- 




1 metre 


+-2 




Oscillator 


wave 

Technique) 


Hertz 1\m 




3 


Shert'Wive 




Wireless 
Methods 




100 metres 


4 


Herteian 








10 «M 


B 


WH 












7 


Ne upper 
limit 










B 














reached are 



due to 

between 



lone infra-red rays. The: limits 
Lebedew, Lampa Sir J. C. Bose (4 mm.) and the 
dect^magneti/ waves and beat waves has been completely fan* 
hv the Nichols and Tear, Arcadiewa, and others. Duiii^ 

World War II the technique ol prodm tion of waves of lengtn ironi a 
few centimetres lias undergone a . te revolution 

. . ,>f magnetron and klystron tubes used toi Rat 
The ultraviolet part (rays shorter than the last visible = 3800) 
was d i sc IT. Ritter in 1802. lie round I .notogr aphic 

plate was affected even beyond the visible limit. < f*« 

usually give lines as far as A = 3400, after which either uvioi glass 
flimit A = 2800) or quartz prisms should be used, At about A — <-> 
the gelatine used in the photographic plates begins to abs , 
Schumann was the first to prepare plates without gelatine and open 
up what is called the Schumann region. Below A== I860 A. U. qua 
begins to absorb heavily, and Schumann therefore used fluontr- 
Bpectroscopea. Beyond A = 1400, air absorbs heavily, and the whole 
ope. ! producing light and photographing the spectrum must be 

done in he pioneer in this field was 1 h Lyman o Han i 

W hc ; le to photograph Hues as far as A = 600. In this region 

-, ra1 ; i be used for producing the spectrum. 

X- f ; shown by M. Lane in 1912 to be light waves of 

remely short wavelength 10— « cm., w, about 1000 times shorter 
than ,-, light. He used a crystal as diffraction grating, lhe 

shortest. X-rays measured by the crystal method has the wave-length 
I U. -/-rays, obtained in radioactive disintegration of the nuclei 
oL atoms, are still shorter, viz., from HH> to 10-«> cm, in wavelength. 
The gap between the Lyman region (600 A. U.) and the soft 
X-ray region (aboul 20 A. U.) lias been gradually by the works 

of Millikan and Bowen, Compton, Thibaud and others. Mlllikan and 
Rowen used ordinary vacuum spectrographs with diffraction g 
Their source of light was condensed vacuum spark. For ' wavelet i 
in this region, crystal gratings have then- spaces too small, while tfo 
ruled gratings have their spaces too large. Conipton instituted 
method of obtaining spectra in this region with the aid o 
grating at glancing angle (see M. X. Saba and X. K. Saha, Treatise 
.Modern Physics, Vol. I, p. 273). 

Since radiant heat and light are identical, all the laws and 
theorems of Optics and Spectroscopy can be applied i udy 

of m. Rut in this chapter, we shall deal with the subject 

only as far as it is connected with heat. We shall first enter into 
a preliminary discussion regarding the passage of radiation through 
matter, 

5, FnndamentaJ Radiation Procewe*.— Every hot body emits r; 
tion from its surface which depends upon the nature of the surJ 
its size and its temperature. This is known as emisai 
When the emitted radiation falls on matter, a part i\ reflet 



286 



RADIATION 



[CHAP. 



and transmission are connected by the relation absorption 

where r — fraction of total energy reflected 

a — " " » » absorbed 

i . _ 
— M « » « transmitted. 

all ^ 'f CB n ° r t S nsm ^ the % ht ««<* falf/onTTC ,' 
all, and hence appears black. But till uerfectlv hkrfr ui - " 

6. Theory of Exchanges (Prevost— 17ft2l _1>»tW t^ i*oo n j 
regarding emission of radiant enefgy Ze T^ „ ui -f ^w*? 

t a hlocfc of ice, we feel a senJrinn S ? ™ n i Wwn * e stLind 
being at about W lose mLI ~ v C .V M beCTOM om b0 <*3 
* Ice, which is a a SS w5. ? iadaiVm . than il rece " 

phenomena. q * al dnd ma >' ** a PP lie ^ to all si,, 

V>teZ^ t ^ ( ?£p^ d ™ «*! ^en it ^ at the absolute 

becomes necessar tteate hnL it I - ■ ^perature, it 

temperature and the nSM^Ste Tth£ 2?" ^ A 
confine our attention to the *>n> t *t ln ™ S€Ctlon we shall 

stance and the tm'roundin^ V,\ *«&****, both of the sub- 

The experiments can be ^^p^SeVV S^ 



XI. 



LAWS OF COOLING 



bulbed mercury thermometer and enclosing the bulb in an evaa 
flask, the walls of which are blackened inside and are further ma - 
tained at a fixed temperature by means of a suitable bath. The 
rate of fall of temperature of the thermometer gives the rate of 

cooling. It was found that if we plot the rate of cooling - -^- as 

against die excess of temperature as abscissa, the curve is a 
: ight line, or in other words, 

-£ -H0-4) « 

■ B n is the temperature of the enclosur he rate of cooling 

is proportional to the excess of temperature of the substance above 
the surroundings. This is Newton's law of cooling. The law holds- 
when the temperature difference is not large. Even the earlier 
experiments showed considerable deviations from the law when the 
temperature difference exceeded 40 a C. 

In order to find out a law which wiH hold for all differences ol 
temp;: of experiments were performed by 

Dulong and Petit who found that their results were given by a formula 
of the form 

_d0 
' tit 



= k{aO —a&o), 



(2) 



where > instant and k depends upon the nature oil the emitting 

e, and 0,6 O denote respectively the temperatures of the emitting 
. jd the surroundings. 
.Stefan* showed later that the results of Dulong and Petit 
be represented by the equation 



-£-• <*•-*». 



(3) 



where a is a universal constant, and T and T the absolute tempera- 

tares oi the body and its surroundings ; or in other words, the emissive 

t of a substance varies as the fourth power of the ah 

erature. This is Stefan's law and can be deduced frort] theo 

. 1 considerations (vide sec, 26). It is therefore the correct law 

of cooling for black bodies, 

Newton's law of cooling can be easily deduced from Stef; 
law when the temperature difference is small. Thus if the body 
at temperature T -f- 87* is placed in an enclosure at temperature T°K, 
the rate of loss of heat, per second by unit area of the body is 

a (T+8T)*-aT* 

- „T* / 1+"^ ) '- ^r 1 = ±vTHT 

neglecting higher powers of ST. It is thus proportional Co the tem- 
t.ure difference 87'. 

♦Josef Stefan (1835—1893) was Professor oi Physics in Vienna. He dis- 
covered the law known by his tuutie, 




288 



KADIATIOK 



[CHAP, 



Generally for rough work in the laboratory ! law of 

cooling is employed Eor correcting for heat loss even when the body 

suspended in vacuum. As shown in Chap* VIII, § 30, if 

cooling takes place by natural convection in air, Newton's law 

not hold. In accurate work tct rate of cooling should be 

irved and not computed. 

Exercise*— A, body cools from 50°G to 40°C. in 5 minutes when 
trroundings are maintained at 20°C. What, will he its tempera- 
ture after a further 5 minutes? Assume Newton's law of cooling 
to hold. 



Nov,- for .: ^ 0, = 50 

t=W t 9 = X 

.*, C - lag, 30 j £ = - 



,-. log (0-20) =-kt + C. 

5, 

(to be determined) . 
1 . 20 

5 ' ''" 30 



•'■ log4 "itr = 2 logj M whcncc 6 *=M3*s] 

8. Emissive Power of Different Substances — Preliminary Experiments. 

—Having n experimental study of how the emissive power 

with i let us find how the 

with the nature of the surface ig body, is pur- 

Leslie : ted with i i while 

I with the sul to be 

n both 1 to fall 

itio of 
led by Lite thermopile in Lh gives, 

In this 
missivitii mined, 

Lesl 

9. Reflecting Power. — T.I tig power of difl mb- 

found by allowing the radiation from a 
Leslie cube to n Be in surface and the refle< 

, Next the thermopile is placed at the 

-surface. The ratio of ns in 

I Meeting powi rbis 
^e wi th I radiation employed, and as earlier workers did not. 

employ monochromatic radiatit i lults are m i ach value. 

10. Diathermancy. — Diachermanq with regard to heat is 
analogous to transparency in the case of light. Subsi which 

allow radiation to pass through them are called diatk 
those [o not, are "called atk\ us. The early experiments 

in this di performed by Melloni and I. Direct 

radiation from a Leslie cube fell on a thermopile produ deflec- 

tion ; next the experimental substance was introduced and the deflec- 
tion observed. The ratio of the two deflections gives the diathermancy 



XT. 



RELATIONS BETWEEN THE DIFFERENT RADIATION QUANTITIES 



289 



or transmission coefficient of the substance. Solids, liquids and gases 
were treated in this way. The best diathermanOi are rock- 

salt {Nad), sylvine (KC1) , quartz, fluorite mid certain other crystals, 
diathermanous, but water vapour and carbon 
dioxide show marked absorption. 

11. Absorption. — For studying the absor] lids Melloni 
took a copper disc and coated one face with lamp-black and the other 
Tare with the experimental substance. The coppei dm was placed 

v,„ the L' abe and the thermopile, the Ian -black surface 

the thermopile. The plate will assume mperature 

T. ' al radiated per unit area of the plate 

H= (E + B)T. 
where E and E denote the emissivity of lampblack and of the 
substance. Now this must equal the radiation which it absorbs. 
The thermopile reading gi*es ET and therefore E. Knowing E and 
/,;■■ we g e i //. th< ii.sorbed. In this way the absorptive power 

of different si es could be compared. For gases a sensitive 

arrangement ed, 

terms diathermanous or athermanous lark in scientific 
precision. Every substance ought to be defined, so far as its trans- 
mitthij sorting properties are concerned, with respect to a parti- 

cular wavelength. In the above experiments monochromatic radia- 
were not employed. The whole subject is now studied as a 
branch of physical optics under the head 'dispersion and reflection' 
with which it Is intimately connected. 

12. Relations existing between the different Radiation Quantities.— 
The foregoing experiments however show that at the same tempera- 
ture a lamp-black surface emits the maximum energy while a polished 
surface emits very little energy. It was further found that radiating 
and absorbing powers vary together; that good radiators are good 
absorl I pocr reflectors, while poor radiators are poor absorbers. 
Lamp-bin aque to radiation but allows radiation of very long 
wavelength to pass through it. 

A very simple experiment devised by Ritchie demonstrates 
vividly the relation between emissive and' absorptive powers of a 
body/ A Leslie cube AB, which is a hollow 
metallic vessel and ran be filled with liquid 
at any temperature, is placed between the 
two bulbs of a different; aeter 

(Fig. 8). The face A of the cube and the 
bulb D are coated with lamp-black while the 
face B and the bulb C can be coated with a 
layer of the substance whose emissivity is to 
be' investigated, my powdered cinnabar. By 
filling the cube with a hot liquid the index 
not found to move, a ace the 

amounts of energy received by C and D are equal. If e denotes the 

19 



C 




6. — Ritchie's experi- 
ment 






RADIATION 



[CHAP. 



►f heat emitted by the substance and a its absorptive power, 
lenote the corresponding quantiti. : amp-black, 



i 



e 
E 



a. 



w 



Now <?/£ may he called the coefficient of emission o£ the sub- 
atancej and hence the relation shows that the coefficient of emission 

ual to the absorptive power. It will be seen that the effect of 
temperature upon the coefficient of emission has been neglected here. 
Nevertheless the experiment shows at least qualitatively that the 
coefficient:? of emission and absorption vary in the same manner from 
one substance to another. 

These ideas were further developed and made more precise by 
Kirchhoff and Balfour Stewart. Before proceeding further it is how- 
ever necessary to define the concepts used in Radiation with more 
rigour. We now proceed to do this in the next section. 

13. Fundamental Definitions. — If a body is heated it radiates 
Uons its surface in all directions, which comprises waves of all 

The nature of radiation depends on the physical properties 

of the boi denote by e\dk the amount of radiation measured 

js emitted normally per unit area per unit solid angle per second 

within the wavelengths A and \-\-d\. We shall call e ?i the emissive 

of Che body. arly if dQ\ be the amount of radiant energy 

a the body in the form of radiation (A to A + dk) and a 

absorbed by the body and converted 

is called the absorptive power of the body for these 

to A-f dX). For black bodies a\ =1 for all wavelengths, 

r other substances a. depends on the physical nature of the body. 

14. Kirchhoff's Law.— In sec. IS « Chat emission and 
i vary together. In 1859, Kirchhoff* deduced an important 

which may be stated as follows : — 

%iven temperm ' ratio of the emissive power to the 

''live power is the saute for all substances and is equal to the 
>er of a perfectly black body. 

Though this law was first recognised bv Balfour Stewart, 
the first to deduce it from therm odynaniical principles, 
apply it in all directions. It is therefore usually known as 
KirehhorTs law. 

We have considered here the total emission regardless of wave- 

Li but the same relation holds for each wavelength separately. 

radiations of the same wavelength and the same tempera- 

* Gustav Robert Kirchhoff (1824 — 1887), bora at Kfioigsberg; became Pro- 
fessor of Physics at E and Heidelberg, He h noted for his discoveries in 
spectrum analysis and the radiation law bearing his name. 






XI.] 



APPLICATION OF KIRCHIJOFF S LAW 



291 



turcj the ratio of the emissive and absorptive powers for all bodies 
is the same and equal to tl live power of a perfectly black body. 

This also holds for each plane-polarised component of any ray. 

Kirchhoff's own proof of the law is however very complicated 
and has. so I fallen into disuse. We shall give in sec. 19 a much 

simplified proof of the law. But before doing so let us consider its 
applications. 

15. Applications of Kirchhoff's Law.— The law embodies two 
distinct relations, a qualitative and a quantitative one* Qualitatively, 

it implies that if a body is capable of emitting certain radiations it 
will absorb them when they fall on it. Quantitatively, it signifies that 
the ratio is the same for all bodies. 

Various experimental proofs and observations may be cited in 
support of die qualitative relation. If a piece of decorated china is 
heated in a furnace to about 1000°C and then taken out suddenlv in 
a dark room, the decorations appear much brighter than the white 
china, because these being better absorbers, emit also much greater 
light, If we take a polished metal ball and have a black spot on it 
by coating it with platinum black., then on heating die ball to about 
I000°C and suddenly taking it out in a dark room, it will be found 
that the black spot is shining much more brilliantly than the polished 
surface. Again take die case of a coloured glass. We know that 
green glass looks green because it absorbs red light strongly and 
reflects the green (red and green being complementary colours) . 
Hence when a piece of green glass is heated in a furnace and then 
taken out, it is found to glow with a red light. Similarly a piece of 
red glass is found to glow with green light. A more decisive example 
illustrating selective action is that of erbium oxide, didyrnium oxide, 
etc., which when heated emit certain bright bands in addition to die 
continuous spectrum. If now a solution of these oxides is made and 
continuous light, is passed through it, die very same bands appear in 
absorption. 

1$, Application to Astrophysics. — B e s i d e s these applications 
Kirchhoff's law was in a sense responsible for the birth of two entirely 
new branches of science, viz., Astrophysics (physics of the sun and 
the stars) and Spectroscopy. We shall recount here briefly how these 
developments grew out of Kirchhoff's law, Newton had shown in 
1680 that the sunlight can be decomposed by means of a prism into 
die seven colours of the rainbow, but Fraunhofer who repeated the 
experiment in 1901 with better instruments, found to his surprise, 
that the spectrum was not continuous, but crossed by dark lines. 
Their number is at present known to be 20000, but Fraunhofer noticed 
about 500 of them, and denoted the more prominent bands by the 

letters of the alphabet : — A, B, G, Such a solar absorption 

spectrum is shown in Fig, 9 (C) togeiher with a continuous prismatic 
(A) and continuous grating spectrum (B). 



T 






RADIATION 



CHAP, 



Frau nhofer never understood how these dark lines originated, 
neither did any oi Jus contemporaries. But he realized ilu i: great 
red them and catalogued their wavelengths. He 
examined the light from stai . and showed that then .p 
also crossed by dark lines just as tn the case of the sun. A tj 
stellar spectrum is shown in Fig. 9 (E) . 

In : antime, however, other sources of light, were examined 

by the spectres id some knowledge was obtained of their spectra 

and that a glowing solid gives a continuous spectrum, hut 
a flame tinged with NaCl pair of intense yellow Hues 

dark background. It was also found that, if an i discharge was 

I through a glass tube containing gas at low pressure, a large 
number of emission lines were obtained on a dark background, For 
hydrogen these are shown in Fig, 9 (D) . 

But still the dark lines of Fraunhofer remained unexplained. 

Some physicists, notably Ffeeau, o I that if the spectrum or 

the sun > - Id with the spectrum from a sodium 

flame, the yellow lines appear in the same place as the D-band of 

the FraunhoFej rum. Similar is the case with the hydrogen 

rum. in Fig. 9(C) ai I we see the emission spectrum of 

hydro by side with the Fraunhofer spectrum. It is clearly 

m some of the dark lines in the latter occur in the same place 

bright lines in the former. 

Th Iven by KirchhoS not only completely solved 

; : r-reacning and extremely fruitful in its con- 
ned that the central body of the sun consists 
ich emits a continuous spectrum without I 
i lighl has to pass through a cooler atmosphere sur- 
mass. In this atmosphere all the elements like 

-c etc, :n ]i, in the gaseous form in addition to 

' :. ::tc. 

W: : ri how the early workers studied the phenomenon 

of radiation with simple apparatus and arrived at some very 
general laws. Every substance when heated emits radiation, Le tJ 
light. Every substance has again got the power of absorbing light. 
Kirdihoif arrived at the same law from thermodynamic reasoning 

19) and applied i(- to explain the dark lines. Sodium can 
the D-Iines when it is excited; hence when white light falls on it, it, 
can abs ; ' also the same light, and allows other light to pass through 
it unmolested. The gases in the outer cooler mantle round th 
therefore deprive the continuous spectrum From the central mass, of 
the lines they then ran emit, and give rise to the black lines. 

The D-bands therefore prove that there is sodium in the sun's at- 
mosphere. Similarly, the other dark lines testify to the presen 
their respective elements in the atmosphere of the sun. 

The correctness of Kirch hoi i matJon. is seen farther from 

flash spectrum results. We have supposed that 'the atmosphere of 



To face p. 292 



CONTINUOUS PRISMATIC 





CONTINUOUS GRATING 








SOLAR ABSORPTION GRATING 

FED C B B a 


A 


(oTT 






rafw , , , | 


(D)l 









HYDROGEN BRIGHT LINE 
SPECTRUM OF ^-AURIGA (G r , TYPE) 



(E) 



■ 



COMPARISON OF FLASH SPECTRUM 

(1905) WITH ROWLAND ATLAS 

(Flash photograph enlarged six times and Atlas reduced five limes). 




(F) 

m ii ii Aiiiiiiiliiiiiiiiiiiiiiii 




Fig. 9 
Figs. (A), (B). {€), (D) have been reproduced from Know] ton, Physic 
flege Students, (E) from Muller-Pouilicts. Lehrbuch der Physik, Vol. V, 
part 2, and (F) from Handbuch der Astrophysik-, Vol. IV. 



I 




XI, 



APPLICATION OF THERMODYNAMICS TO RADIATION 






the sub contains Na. Now if we could observe the spectrum of the 

atmosphere apart from that of the central glowing core (the photo- 
sphere}, tlu: line uid appear I We cannot ordinarily do so 
because the solar atmosphere is so thin that we cannot cover up the 
disc properly, leaving the atmosphere bare for our observation ; 
further the scattered sky-light completely obscures the spectrum of 

the pari outside the dis during a total solar eclipse, the 

solar disc i ■ completely coveted for a short time by the disc of the 
moon, and skylight is also reduced by the moon's shadow. To anyone 
observing the sun through a spectroscope, the solar atmosphere will 
be laid bare al the time of totality and the dark lines will flash out 
as bright lines. This was Ei iii d actually to be the case by Young of 
Princeton in 1872, Fig. 9 (F) shows a Hash spectrum of 1905 eclipse 
which is placed side by side with the Rowland Atlas of Fraunhofer 
spectrum for the sake of comparison. The lines of the flash spectrum 
are thus found to be due to the elements in the sun's atmosphere. 

But Kii Hi ho |]"s disco voi}' is of much more far-reaehing import- 
ance than the mere success in explaining the. Fraunhofer bands would 
indicate. It clearly asserted for the first time, that ev& rent 

type of when it is properly excited, emits light of defcni 

wavelength which is characteristic of the atom. Just as a man is 
known by his voice, ot a musical instrument by the quality of its 
note, so each atom can be recognised by the particular lines it emits. 
Thus was bom the subject of Spectrum Analysis, which aims at 
identifying elements b) their characteristic lines, and forty new de- 
nts 'were added with its aid to the list already known. The 
different aton rded as so many different types of instruments, 

each capable of producing its own characteristic aetherial music. 

APPLICATION OF THERMODYNAMICS TO RADIATION 

17. Temperature Radiation. — Since material substances at alj 
temperatures are found to emit radiation, it becomes possible to 
apply the laws of thermodynamics to the problems of temperature 
radiation. The expression "Temperature Radiation" should I 
clearly understood, for matter can be made to emit radiant energy 
in many way. othei than by heating, e.g., by passing an I '• trical dis- 
charge through it when in a gaseous state, by phosphorescence, fluores- 
cence, or by chemical action as in flames. But Kbchh oil's law holds 
only for temperature radiation. For radiation produced by other 
methods the law cannot be applied,! 

18. Exchange of Energy between Radiation and Matter in a Hollow 
Enclosure. — Let us suppose that there is an enclosure with its walls, 
which are impervious to external radiation, maintained at a cons- 
tant temperature. We shall study what takes place when we place 
substances having widely different physical properties within this 
enclosure. 

In the first place the whole space is fdled with radiation which 
is being emitted by the walls. This radiation arises out of the heat 







294 



RADIATION 









* M??^ b ^ zi "^ «W««W*y squire the same temperature 
the walk* ihu can be proved by the method of nducfja 
***. .tor suppose that inV equilibrium state the tem 
evSkLJi 11 a £a™°t engine w*T ^ used to a heat 

ESTSS T, ' W ? lls t0 A unU1 A J,ad die samc temperature as 
the walla. During the process a certain amount of heat will 

SmESJ 1 ^ - But ^ in A , would be reduced automatically i 

Si r /^J*™""- ;f according to our assumption, this is the 
.table state o/ affairs. Hence again the difference of at ure can 

he l hcdv J^ f* C0nVei ' SiOn ° f hCat f ° W ° rL We • **™ 

wk ' 7** ° 38SU ^ a temperature different from that of the 
g. we have at our disposal a means of continuous!, . ting 
the heat of a single body to work without mamtaininj 

conclude tfut ri materia] bodk: | J n the enclosure woi 

mately assume the temperature of the walls. 

fa vi^V'Vr' fPPosf that a body of hti [position 

£P '-Inn the enclosure. Then the different parts of iL'body" 

toe different emissive and absorptive . The total euro 

ab> " * equal to tile energ , S\iSfS 

temperature reman, llt [n the equilibrium , Z f t j£ 

T" »m PrTvost's 

2J « Allows the 

* - « ****** is changed, And as the d'fS 

l" 1 ' Cerent coefficients of , ,,,. rhl ca"i 

™ h mica! in qua, ..,., L cpiab 

of the enclosure or of any body plao [e it. T.ei L £ 

we have two enclosures A and B having walls of differs X w 

^n?^ion u " fca ; ing f, h - cach *** tht " ?*s* 

?h™ i?V xt of L WaveIe "St h lymg between A and A + o'X to pass 
through it. Now the walls of both A and B are mainlined at Z 
™ jure T. Tf in the steady state the intenritv of radt at o^ 

ms thrown ?£ «f ater * A than in B. some fadiation will 

Thi Iv« and decrease in A. The screen is then closed 

The excess radiation will be absorbed by the walls of R and raise to 

temperature to r, while the temperature of thesis of TwTfaTL 






deduction of kirciuioff's law 






Now a Carnot engine may be worked between these two temperatures 
ding :l certain amount of work and lowering thereby ipe- 

■ '■;' li and increasing that of A till the two are brought to the 
irature. The process can be repeated and thus we 

Duntol 'work while the entire system : cm 

colder. Thus in effect we are getting work indefinite!) by 
usii Meat of a single body. This is Impossible by the second 

lynamics. Hence the intensity of radiatioi 

the same in the two enclosures, i.e., the quality 
. of radiation depends only upon the temperature and on 
ii i thing else. 

If we now place a black body inside the enclosure, it will emit 
.in energy of the same quality and intensity as it absorbs. Hence it 
follows that the radiation inside the enclosure is identical in ev 
the radiation emitted by a black body at the same M 
perature as the walls ol the enclosure. These concha were 

rchhofE in I 

19. Deduction of Kirchhoifs Law.— Suppose we 5 

some sub ive and 

of wavelength lying between A and 
iid ax wly. We have alread a in the 

don that the amount of radiation dQ falling on the sub- 
die wall i depend upon the nature < hape 
of the walls. Of this a portion ot A dQ is I ■ nrbed b) 

remainder (1 -a A ) dQ, is reflected or transmitted. Further 
the su radiation equal to e^dX by virtue 

perature. Equating the energy absorbed to the energy emi 
■ 

e x d\ = ax<i(l 

In the case of a perfectly black body of emissivity have 

(since a\ = I) , 



Combining (5) and (6) we get 

- F 

: — r. ; 



• (6) 



(7) 



■;■ ratur the ratio of the emissive power to the dbsarp- 
ibstance is constant and equal to the emissive p 
of a perfectly black body. This is the thermodynamic proof oi 
Kirchhoftfs law. We have proved the law here for bodies inside the 

enclosure. Now since tire emissive and absorptive powei I only 

upon the physical nature of the body and ijui upon its sin i 

it follow-s that the law will hold for all bodies under all conditions Eot 

pure temperature radiation. 



II ■ 



29G 



KA IU ATI ON 



[CHAP. 



20. The Black Body,— The considerations put forward in sec. 
enable us to design a perfectly black body lor experimental purposes. 
seen that it an enclosure be maintained at a constant tern- 
are it becomes filled with radiation characteristic of a perfectly 
black body. If we now make a small hole in the wall and examine 
the radiation coming- out of it, this diffuse radiation will be identical 
with radiation from, a perfectly black emissive surface. The smaller 
the hole, the more completely black the emitted radiation is. Thus 
a correction has to be applied for the lack of blackness due to the 
finite size of the hole. This h due to the fact that some of the 
radiation coming from the wall is able to escape out and the state of 
thermodynamic equilibrium as postulated in section IS does not hold. 
This is almost completely avoided in the particular type of black body 
due to Fery (Fig. 11, p,' 297) . So we see that the uniformly heated 
enclosure behaves as a olack body as regards emission and if we make 
a small hole in it, the radiation 'coming out of it will be very nearly 
blackbody radiation. 

Again such an enclosure behaves as a perfectly black body* 

towards incident radiation also. For anv Tay passing 1 into the hole 

will be reflected internally within the enclosure and will be unable to 

escape outside, This may be further improved by blackening the 

Hence the enclosure is a perfect absorber and behaves as a 

ectly black body. 

Though Kirchhoff h cd in 1S5S that the radiation inside 




3%ffi g ^a s f^,a3g-ESBSafefli j ^ 




Rheostat 

Fig. 10.— Black body of Wien. 

a uniformly heated enclosure is perfectly black it was long afterwards 
in 1895 that Wien and Lunmier utilised this conception to obtain a 
black body for experimental purposes, m 

The black body of Wien consists of a hollow cylindrical metal! t 
chamber C (Fig, 10) blackened-]- inside and made of brass or platinum, 

* Kirchhoff defines a black body as one "which has the property oi allowing 
all incident rays t enter without refk-etii u ;iii * them t:> leave again. 

See Planck, Wormcstrnhhing. 

| The walls however need not be black. Blackening merely enables the 
equilibrium state tu be reached quickly. 



XI,] RADIOME1 297 

depending upon the temperature that it has to stand. The cylinder 
ated by an electric current passing through thin platinum foil as 
indicated bv thick dashes. The radiation then passes throng 
number of limiting diaphragms and emerges out of the hole O. The 
cylinder is. surrounded by concentric porcelain tubes- The tempera- 
ture of the black body is given by the thernio-element T. This is 
the type of black body now commonly used. 

Another type due to Fery is shown in Fig-. 11. Note the conical 
projection P opposite the hole O. This is to avoid direct radiation 
from the surface opposite the hole which would otherwise make the 
body not perfectly black, 

A striking property of such an enclosure is that if we place any 
substance inside it, the radiation 
emitted from it is also black and in- 
dependent of the nature of the body. 
Thus all bodies inside the enclos 
lose their distinctive properties. For 
a mirror placed inside will reflect the 
black radiation from the wall and 
hence the emitted radiation is black. 
Any substance if it absorbs any radia- 
tion transmitted from behind must 
emit the same radiation in conse- 
quence of Kirchhoff *s law and the 
total radiation leaving it must be- 
come identical with that from the walls i.e., of a black body. 




Fi.cr. 11- — Black body of Fery. 



RADIOMETERS 



21. Sensitiveness of the Thermopile. — We shall now describe the 

instruments which are used for the measurement, of radiation. The 
differential air thermometer, which was employed by Leslie and the 
early workers, has already been described in sec. 2, but is now only 
of historical interest. 

Among the modern instruments, the thermopile has been des- 
cribed in sec, 2,. Special care however has to be taken in order to 
make it sensitive as we have sometimes to measure very small amounts 
of energy. In the design of thermopiles, the following 1 considerations 
have to be borne in mind : — (1) metals used should give large thermo- 
F..M.F. ; (2) junctions should be as thin as possible ; (3) connecting 
wires should be thin so as to minimise the loss of heat, by conduction ; 
(4) the junction should be coated with lampblack so that all the heat 
falling on it may be absorbed ; (5) it should be mounted in vacuum, 
so that there is no loss of heat by convection, and the deflections 
remain steady ; a rocksalt window is provided to let in the incident 
radiru 

The sensitiveness also depends upon the number of thcrmo- 
junctions but this cannot be increased indefinitely as the external 



298 



RADIATION 



CHAP. 



resistance will increase. The best procedure is to have the piles as- 
light as possible and to choose a number so that the total resistance 
of the thermopile is equal to the galvanometer resistance. 

The galvanometer to which the thermopile is connected should 
be o£ low resistance type with high voltage sensitiveness. For ordinary 
work the .suspended coil type galvanometers are generally employed 
but are not sufficiently sensitive on account of their high resistance. 
For accurate work the suspended magnet (astatic) type of galvano- 
generally employed, namely, the Broca, the Paschen and' 
the Thomson galvanometers. These are however very much suscep- 
tible to external magnetic disturbances and can be successfully used ' 
only by skilled workers. 

The Linear Thermopile — The extreme sensitiveness of the galva- 
nometers mentioned above enables us to work with only a few i 
of thermo-elements. The hot junctions are all arranged, in a vertical 
line. The wires are very fine and are wound" 
on n small frame. This is called the linear 
thermopile and is used for investigating the 
lines of the infra-red spectrum, 

22. Crookes' Radiometer, — This consists 

of a number of thin vertical vanes of mien sus- 
pended at the ends of a light aluminium rod 
r inside an evacuated glass vessel (Fig, 12). 
Two such rods fixed at right angles are shown 
in the figure. They are suspended in such a 
can rotate about the vertical 
- a. The outer face of the vanes is coated 
b lampblack, while the inner faces are left 
clear. When radiation (thermal or light) 
falls on the blackened face the vanes begin 
to revolve in such a direction that the black- 
ed fare continually recedes away from the 
source of radiation. 

The cause of this motion is easily under- 
stood. The blackened face absorbs the Inci- 
dent radiation and thereby its temperature 
is raised, while the clear face remains at a 
lower temperature. The molecules bombard- 
ing the blackened face therefore become more 
hen ted than the molecules bombarding the 
other side of the vane and consequently they exert greater pressure 
on the vanes. The result is that there is a net effective force repelling 
the vanes from the incident radiation. This is known as the radio- 
meter effect, and is essentially due to the presence of molecules. It is 
obvious that the velocity of revolution will be a measure of the in- 
tensity of radiation. 

By suspending the vanes by means of a quartz fibre as in galvano- 
meters we can measure the intensity of radiation if we observe the 




. 12.— Crookes' 
radiometer. 



W 



xi.j 



THE BOLOMETER 






i .j Wi a radiometer has been employed 

steady deflection f^ft**;*^ ™X*#J of radiation. 

pUtinuSr strips or wires when .heaKd. ,,. thc ^native- 

P Tire WtiveaeB rf thebolo *«e r d^penc... ^ ^ fc 

ness of the galvanome er, W "° ™\ e the rise in .emperaM* and 





Figs. 13 and 14,-The Surface Bolometer. 

(2 ) linear bolometer to neueftt of distribution of energy in the 

spectrum of a black body. f evceedinsrlv thin strips of 

method of constructing such a 
thin conductor is as follows :— A 
sheet of platinum is welded to a 
thick sheet of silver and thc com- 
posite sheet is rolled. The sheet 
is then punched out as shown in 
Ficr 13, and attached to a hollow 
frame of slate. The ^lve 
dissolved off in nitric acid, tne 
end joints being protected by a 
coating o£ varnish. The strips are 
then coated with platinum black. 
Fitr 14, shows a front view ot 
such a grid. A grid so const 
ed has a resistance of about W 

° hm For experimental purposes n^- 

fe^t e^'otte^'ntected in the form of a Wheatstone 




800 



RADIATION 




bridge. The method of connecting the grid is shown in Fig, 15. The 
grids 1 and 3 are in opposite arms of the Wheatstone bridge and are 
so arranged that the strips in 3 receive the radiation passing between 
trips in I and so die effect is doubled, 2 and 4 are similarly 
arranged but a] cted from radiation. The whole is enclosed in 

a box, In the absence of incident radiation the galvanometer shows 
no deflect ion. When radiation is incident on grids 1 and 3 defection 
i] oduced, 

In the linear bolometer a single narrow and thin strip of plati- 
num is as 

24. The Radiomicroraeter. — ThJ vented by Boys and is 

essentially a thermocouple without an external nneter. A 

single loop c of fine copper or silver wire is suspended (Fig. 16'i 
between the poles of a strong permanent magnet NS as in a suspended 
coil galvan To the lower ends of the copper wire two thin 

bars of antimony So and bismuth Bi 
are attached, and the lower ends of 
these are again attached to a thin disc 
d or narrow strip of blackened copper. 
To the upper end of the copper lo 
is attached ;i thin h g carrying 

a light galvanometer mirror m few 
measuring deflection, the glass rod it. 
attached to a fine quartz 
pension q. The whole system is ex- 
uely light. Radiation, falling hori- 
i tally on the copper disc, beats the 
junction of antimony and bismuth and 
an current flows through the 

copper coil causing a deflection which 
depends upon the intensity of the 
energy re tivcd by the copper disc, To 
prevent disturbance due to diamagnet- 
ism of bismuth, Che rods of the thermo- 
couple are surrounded by a mass of 
soft' iron. The whole suspended sys- 
tem is enclosed in brass (shown shaded 
in the diagram), 

25. Pressure of Radiation. — As ra- 
diation has been shown to be identical 

with light, it possess::-- :il the properties 

which are ascribed to light. One of 

the properties of light most important for our present purpose is 

it exerts a small but finite pressure on surfaces on which it is 

incident. This had been suspected by philosophers for a long time 

i he days of Kepler, who observed that as the comets approach 

•Reproduced from "The Theory of Heat" by Preston by the kind pcrmb 

> i .',',._■ <-:-■ j, I; .T-ill.i:-: & Cb. 




Fig. 16,* — Boys" Ri 
micrometer. 



PRESSURE OJ KAHM ' 



SOI 



the sun, the tail of the comet continuously veers round, so as 
to be >pposate the sun (Fig. 17). This he tried to explain 

on the assumption that light exerts pressure on all material bodies 

hich it is incident, but th< Lre increases in importance only 

the size of the particle is reduced. The die sun 

cometary matter, either dust particles or atoms, which are then 
repelled by light-pressure and thus lorm die tail. Though painstak- 
ing experimental investigations failed to show the existence ot pressure 

"it., Maxwell propounded in 1870 his Electromagnetic -Theory of 
Light, lowed that even on this theory Light should exert a 

ire but this is very small, being equal to the intensity divided 
by the velocity of light or Co the I jy-density. Calculation shows 
that the pressure due to sunlight is equal to 4-5 ;spercm* a 







Fig, 17,— Tail ot the comet' 

Bartoli also showed from thermodynamic considerations that 
radiation should exert some Finite pressure. The pressure of radiation 
is however so small that for a long time it battled all attempts to 
measure it. The difficulties were overcome only in 1900 when 
Lebedew, ant! a little later, Nichols and Hull demonstrated c 
mentally its existence and were able to measure it. They confirmed 
the theoretical conclusion that pressure is equal to the energy density 
of radiation. 

When the radiation is diffuse, it can be shown that pressure is 
equal to one-third the energy density of radiation. 

The fact that radiation exerts a finite pressure, however small 
it may be, is of great importance in the theory of black-body radiation. 
It shows tha ick radiation is just like a gas, for it exerts pressure 



%\[>. 



RADIATION 



CHAP. 






EXPLRIMENTAL VERIFICATION OK STEFAN S LAW 



SOS 



and possesses energy. In fact we can regard the black radiation as a 
thermodynamic system and calculate its energy and entropy, and apply 
the thermodynamic laws and formula:. We shall make use of these 
i in the next section. 

26. Total Radiation from a Black Body,— The Stefan-Boltzmann 
L aWt — As already mentioned in sec. 7, J. Stefan, in 1 870., deduced 
kallv from the experimental data of Dulong and Petit that the 
total radiation from any heated body is proportional to the fourth 
power of its absolute temperature. In 1884 Boltzmann gave a theore- 
tical proof of the law based on thermodynamical considerations. lie 
red that the law applies strictly to emission from a black body. 
The law is therefore generally known as the Stefan-Boltzmann law, 
and may he formally' enunciated as follows : — If a black body at 
pemture T be surrounded by another black body at 
lute temperature T the amount of energy E lost per second per 
of the former is 

E = *{T*-Tf) t .... (8) 
where <r is called the Stefan's constant. 

For proving this law, we consider radiation in a black-body 
chamber and apply the thermodynamical laws to the radiation as 
mentioned in the previous section! Let u denote the energy density 
of radiation inside the enclosure, V its volume and p the pressure of 
ion. Then both U and p are simply functions of the absolute 
erature T. We h ther the total energy U of radiation 

equal to uV. Applying* equation (61) p. 267 we get 

&-t. ■ ■ ■ <9) 



and 



1 = 

It 

tdU\ A u 



(10) 



or 



or 






du _ . 
tt 



the radiation is diffuse (sec 25) . Hence equation (9) reduces to 

T du u 

3 ar "" T 

,ir 

T * 

u = aT*, [11) 

where a is a constant, independent of the properties of the body. 
ce the total energy lost on one side by emission will also be pro- 
portional to the fourth power of the absolute temperature. That is 
the Stefan-Boltzmann law. 

27. Experimental Verification of Stefan's Law.— The law was sub- 
jected to experimental test by various investigators, Lummer and 

♦Boltzmann deduced the law by imagining the_ radiation to perform a Camat 
cycle. i :, -'d the conceptions of Bnrtoli. This is however unnecessary here 

for we have shown black radiation to be analogous to a gas and cm therefore 
:1 directly to apply the general thermodynatnieal laws to radiation, 






Pringsheirn investigated the emission from a black body over the 
range of temperatures lOCPC to 1260*0 and found the law to hold 
true within experimental errors, We give below a brief account of 
their apparatus and arrangement. 

A is a hollow vessel containing boiling water (Fig. 18) which 
acts as a standard source of radiation for calibrating the bolometer 
from time to time. The black body C employed for the ran; 
temperatures 2Q0°C to 600 °C consisted of a hollow copper sphere 
blackened inside with platinum-black, and placed in a' bath of a 
mixture of sodium and potassium nitrates which melts at 2I9°C. 
This salt bath could be maintained at any desired temperature. 
The temperature could be measured with a t henna-element T. 




18. — Lummer and Pringshcim's- apparatus for verify trig Stefan's law. 

For temperatures between 900 'C and 1300 Q C the black body 
shown in Fig. 19 was employed. D is an iron cylinder coated inside 
with platinum-black and en- 



closed in a double- walled gas 
furnace. The temperature 
inside the iron cylinder was 
obtained by a therm o-e le- 
nient enclosed in a porcelain 
tube passing through the 
furnace. 

The measuring instru- 
ment shown at B was the 
surface bolometer of Lummer 
and Kurlbaum. A descrip- 
tion of this as well as the method of connecting it has been already 
given in sec. 23. Besides there are a number of water-cooled shutters 
so that the radiation can be stopped or allowed to fall at will. 

The procedure adopted was as follows: — The hath was heated 
up to the desired temperature and maintained steady, and then the 




Fir;. 19.— Bl: I-: , for 9£»* to 13D0 B C. 



304 



RADIATION 



CHAP 



shutter was raised to allow radiation from C to fall on the bolo- 
meter, and the maximum deflection registered by the galvanometer 
noted. The bolometer was kept at different distances from the black 
v and the inverse square law verified- Next observations were 
n with the black body at different temperatures. The observations 
were all reduced to a common arbitrary unit depending on the radia- 
tion from the black body A at 100°C. and kept at a dis ■ of 63$ 
. Tf d represents the deflection oF the galvanometer needle, 
T the absolute temperature of the black body, 290° the temperature 
of the shutter protecting the bolometer, then 

d =* *(r*-290*) 3 .... (12) 
from Stefan's law. The coefficient a was found to be constant. 
Hence the truth of Stefan's law is established. 

DISTRIBUTION OF ENERGY IN THE SPECTRUM 
OF A BLACK BODY 

28. Laws of Distribution of Energy in Blackbody Spectrum.— 

From a study of the colour assumed by bodies when their D lire 

is gradually raised (see p. 281) it will be obvious that as the tem- 
perature of a body is raised, the colour emitted by it becomes rich 
in waves of shorter wavelength, Tn fact the wavelength for which 
the intensity of emission is maximum shifts towards the shorter 
wavelength 'side as the temperature is raised. These results were 
also arrived at. by Wien in 1893 from thermodynamic considerations 
n inside a hollow reflecting chamber. He showed 
thai contained in the spectral region 

included within the wave lengths A and A -+- dX emitted by a black 
at temperature T is of the form 

E^X=^f(XT)dX t ;i:-i 

and further if X m denotes the wavelength corresponding to die 
maximum emission of energy and E m the maximum energy emitted., 

vr-*, (i4) 

and E nl T~ & =B, (15) 

where B and b are constants. In other words, if the temperature oi 
radiation is altered, the wavelength of maximum emission is altered 
in an inverse ratio. Equation (13) is known as Wien's Displacement 
law. 

A further step forward in developing the theory was taken by 
Planck who showed from very novel considerations (which developed 
into the modern quantum theory) that the energy density of radiation 
inside the enclosure is given by 

V^isp^r* < I6 > 

where C t = 8-77/* c/A s and c^— ch/k = 4-96 b approximately. This is 



EXPERIMENTAL. STUDY OF BLACK-BODY l 



30! 



K64 



90 



.known as Planch's lam. We shall 310-..- consider how the spectra can 
be obtained experimentally and the foregoing results verified, 

29. Experimental Study of the Black-body Spectrum. — The first 
aatic study of the infra- lectrum was undertaken by La 

who illuminated the slit of a a is of sunlight and 

produced the spectrum by a prism of rocksalt. The rays were fen 

•. I lens on a bolometer which was arranged in n V me bridge 

adjusted for no deflection, 

he spectrum 1 
be produced by a Ro 

grating but on account of 
the considerable Overlapp- 
ing of spectra o£ different m \ 
orclers prism 1 
generally employed. Lew 
of quartz or fiuorite are m 
generally used. It is how- 
ever better to use a con- ^ 

- for focussing 
the radiation. 

Wien *s displacement 
was subjected to a 

scries of experim tests *••■- 

by Paschen, Lummer and 

fsheim, T< u I; c n : ; and 
Kurlbaum. We give below a 

1 ption of the ex- 

periment and of E5-] 

Lummer and I EngsbJ im 

1 ) . They used radii 

ictrically heated 
carbon tube and ced 

the spectrum by refraction * ; '- 
through a fluorspar prism. 
distribution of energy 
was measured by means I \ 
linear bolometer which was 
enclosed in an air-tight ci 
to diminish the absorption 
due to water vapour and 
carbon dioxide. Corrections 
were applied to convert the 
prismatic spectrum into a 
normal one by 1 means of the 
i. dispersion curve of 
fluorspar. The distribution 
of energy in the spectrum tor various temper: ■ sen 62! C 'K 

and lGiO^'K was obi and curves plotted (Fig, 20) . The ordinates 

• powers and the abscissa: w r avelenglhs. The full 

■■ 



:.. 



Si,- 




1 - ■> S Su, 

Fig. 20- — Distribution of fcjjcrgy in the 
1 - kbody spectrum. 



306 



RADIATION 



CHAP. 



lines denote: the curves obtained experimentally while the dotted 

represent the curves calculated from a semi-empirical law | 
by Wiem The total radiation at a temperature is : •;• the 

area • een the curve and the -v-: varies as T 4 . 

The small patches of shaded area represent the aba 

ide and water vapour. ibsorption of fluorsp 

at G[t where the curves are seen to end abruptly* The wavel 
of maximum emission shifts towards the ' as the temperature 
rises. From these curves the values of E m and X m could be read. 
The experimental data are given in Table 1. 

Ta hie 1<—Experim gj ita I fe rification of Wien's 
Displaceme.! 



Temp. D K 


X m in f/s 


i-'ia 


A m T in 

ii lie ton 
e "ees 


1017 


1646 


1.78 


270.6 


2928 


2246 


1460.4 


2.04 


145.0 


2979 


2184 


1250.0 


2.35 


68.8 


2959 


2176 


1094.5 




34.0 


2966 


21 64 


. 




21.5 


956 


2166 


908, 




13.se 




2208 


723-0 


4.08 


. 


2950 


2166 


621,2 


4.53 


2.026 


2814 


2190 



i — 2940 



2188 



The i -ly the vi ions (14) and i 

the valu ws enable us to determine the 

of any sub'-:' -umed to be a black body) if X„ 

, be found out. The value of 6 is seen to be .294 cm X di 

\ | degree. 
This furni-' with a simple method of determining the tem- 

perature of the heavenly bodies. Thus Cor the moon X^ = 14^, and 
lience 

T= rT . \ i lx . ■=210°K. 
Similarly we can calculate the temperature of the sun (sec, 

PYROME1RY 

39. Gas Pyrometers.- — In Chapter I we have already descri 

hydro :;i thermometer having a platinum- 

um bulb. For temperature above B00°C hydrogen cannot be used 
as the thermometric gas because the platinum bulb is permeable to 
it, hence nitrogen is invariably employed. As regards the material 
of the bulb Jena glass can be used to about 500*0 while quartz 
can stand up to about 1300°C but is attacked by traces at alkali from 






XL] 



RADIATION PYROMETER 



307 



hand etc. Ordinary porcelain is porous and permeable to the theimo- 
L but glazed porcelain was formerly employed up to about 
L000°C ; ' hich however the glazing softens and brea u its 

expansion is not regular. Platinum alloys are now invariably used. 
ln P Xv o< platinum and iridium was employed by Holborn and 

the bulb brittle. Holborn and Day found the alloy of 801 t-20Rh best 

£r the purpose u d a modified Lorn, ol this material to 

mt lfiOO'C. The} performed experiments with great rare and took 

It 25 - ;J L -, •; vor so that their value, are 

ed as standard at high temperatures. The correction however 

becomes enormous at these high temperatures and hence there is 

considerable uncertainty in determining the high temperature hxed 

points with gas thermometer. 

31. Resistance Pyrameters.-The platinum resistance PF°"^er 
has been described in Chapter I. It can be W& to about U 
though the melting point or platinum is 1770°. It used above 1000 
however, the platinum undergoes some change and does not return 
to its initial zero and has to be restandardised. _ The mica insulation 

; , sometimes breaks due to moisture getting inside. 

32. Thermo-electric Pyrometers.— The tbermo-ek ■ roraeters 
have been completely described in Chapter I. It is shown there 
that, for temperatures up to 600°C the base coup es O u-cons tanta n, 

i are the most sensitive. The Pt OOPt-lOUh couple i 
however best for all high temperature work and can be used to about 




A 
mo- 



lOCr couple up to 1500°C when the latter is much more sensitive, 
couple of Tr, QOIr-lORu can be used to about 2100*C while ther 
couples of tungsten and an. alloy of tungsten with molybdenum have 
been used up to 3000 °C. 



RADIATION PYROMETER 
33. Temperature from Radiation Measurements.— In flu 



fore- 




went we can find its temperature. We way cither measure the 
total radiation emitted by the body and deduce the temperature by 
making use of Stefan's law (equation 8). These are called lotal 
Radiation Pyrometer*/ Or we may measure the energy i 
imiti :d in a particular portion of the spectrum and make use of 
Planck's law (equation 16) of distribution of energy in the spectrum. 
These are called Spectral or Optical pyrometers. 

Radiation pyrometers possess the great advantage that they can 
be employed to measure any temperature, however high that may 
be or wherever the object may be. The pyrometer itself has not to 



SOS 



RADIATION 



[CHAP. 

be raised to that temperature nor need it be placed in contact with 
the hot body. Farther there is no extrapolation difficulty as the 
radiation Eormube have been found to hold rigorously for ail tempera* 
lures. But the radiation pyrometer suffers from a serious drawback. 
It can measure accurately the temperature of ijodics only. 

It is, however, generally employed to measure the temperature 
of any hot source. In that case it \ at temperature at which 

a perfectly black body would have the same intensity of emission 
(total or spectral) as the body whose temperature is being measured. 
This temperature is called the . *erature of the 

substance and is conseqi lower in all cases than its actual 

temperature. The greater the departure from perfect blackness the 
greater is the error involved. The lower practical Limit for radiation 
pyrometers is about 600°C, for then the emission from substances 
becomes too small to be measured accurately. Still, however 
certain devices it can be used to measure much lower temperatures 
as that of the moon. 

34. Total Radiation Pyroirtelcrs, — Tery was the first to devise a 
radiation pyrometer based on Stefan's law. These pyrometers are 
merely thermopiles so arranged that the readings are independent of 
rne dflsta itween the hot body and the pyrometer. We shall 

describe Very mirror pyrometer which is typical of this class. Fig. 21 

shows a modern type of 

ToMilUfaaioH j mh Radiation 

incident from the right 
side falls on the concave 
or M which can be 
moved backwards and 
forwards for the purpose 
of focussing the radia- 
tion on the receiver S to 
which the hot junction 
of the thermocouple is 
attached. The cold 
lion of the thermo- 
couple is protected from 
radiation by the tov 
T and is further surrounded by the box B which also contains the 
thermocouple receiver, and a small opening just in front of S. The 
electromotive force developed is read on a mtilivoltmeter connected 
indicated in Fig. 21. The instrument possesses no lag. the steady 
state being reached in about a minute. To enable the observer to 




radiation pyron 




ipening 

being Left at the centre of the mirror to allow the incident radiation 

t. pnss. Now if the image of a straight line formed by the concave 

or does not lie in the plane of die inclined mirrors thev will form 

two images separated by a distance (see Fig, 22) and the line will 



si. 



OPTICAL PYROMETERS 



309 



appear broken when seen through the eyepiece. The concave mirror 

is moved till this relative displacement of the two halves of the linage 
disappears, and then the apparatus becomes adjusted. It will be 
en that so long as the heat image formed by die concave 
mirror is larger than the" hole, the thermocouple measures the intensity 
e heat image ot the total radiation. For if the distance 

of the object is doubled, the amount of radiation failing on the mirror 




r V:l-n 



Fi?, 22. — The focussing device. 



is reduced to one-fourth, but as the area of the image is also simul- 
Lisly reduced to one-fourth the intensity is unaltered. Thus the 
indications of the instrument are independent of the distance. Hence 
in actual use it is essential that the object, whose temperature is to be 
measured, should be sufficiently large and should be placed at not 
very great distance in order thai its image is always bie an the 

aperture in the box which in ordinary instruments is about 1,5 mm, 
in diameter. 

The E.M.F. of the couple in these cases is given by the relation 
V = a(T°-T l ) > .... (17) 

where T is the temperature of the black body to be measured, T Q the 
rature of the receiver S and b a constant which varies from 3,8 
to 4,2 depending upon the instrument. Generally T can be negl 
in comparison with T. This departure from the index value of 4 is 
clue to various ran,:-;, It is for this reason that the pyrometer has 
to be calibrated by actual comparison with a standard thermometer 

j radiation from a blackbody chamber or heated strip. 

3S. Optical Pyrometers, — In these, the intensity of radiation 
from a black body in a small width of the spectrum lying between X 
and A -|- d\ is compared with the intensity of emission of the sine 
colour from a standard lamp. The formulas required for this case can 
be easily worked out by assuming Planck's law. There are two typ , 
of optical pyrometers : — (1) the Disappearing Filament type, (2) the 
Polarising type. We now proceed to describe these instruments. 



310 



RADIATION 



[CHAP, 



XL] 



RADIATION FROM THE SUN 



311 



36. The Disappearing Filament Type.— This type of pyrometer 

trst introduced by Morse in America. It was later inrproi 
HTi and Kurlbaum, and by Mendenhall and . A pyro- 

r of this type is shown in Fig, 23 and is essentially a tele 
ig a lamp at the position usually occupied by the c 
a metal Lube containing tbe filament of a lamp L which is heated 
by the battery B and the current can be adjusted to any amount by 

rying the rheostat R. 
C Radiation from the source 

whose temperature is re- 
quired is focussed by the lens 
on the lamp T, where a 
heat image is formed. The 
lamp is viewed through the 
eyepiece E in front of which 
is pi need a red filter gl 

"ties there are a number 
of limiting rms. In 

periment the 
- looks through E 
and varies the current in the 
till the filament becomes invisible against the image of the 
source. If the current is too strong the filament stands out 

:;e current is too weak, th ' nt looks black. The filter 

g to be done for approximately m 

• ' | . ■ i i I [n - O ' : 

v,' nh -Li pie and then extrapolated by t 

rom the strength of the current required to 
matc l : ! the incident radiation can be 

directly calibrated in degrees. 

37, Polarising Type.— In 1901 Wanner cons;. mother p) 
meter in which the comparison was made by the aid o£ a polarising 




Fig. tfiug Filament 'Pyro:-. 




L, 



Fig. 24, — The Wanner Pyrometer. 




device. Here the ray of a particular colour obtained from the source 
is compared with a ray of the same colour obtained from a standard 



electric lamp. A diagram illustrating the essential parts of the ins- 
trument is £iven in Fig, 24. a, b are two circular ho es arranged 

optical axis of the system. Radiation b 
the ; urce enters the ■■ trough a, while the comparison 

:n is supplied b-. which illuminates the rig 

I the latter directs the light on to b, Both beams 

are rendered parallel by means of an ach \c lens L 1( winch is 

ced ai a distance from J t equal to its focal length, lhe 

Llel beams nre dispersed by the direct vision spectroscope fc> and 

tirough the polarising Rochon prism R which separates 

each beam into two beams polarised in orthogonal planes. B is a 

biprism placed in contact with a second achromatic lei 

f n r, : | ams on the slit D 2 - The biprism produces in 

aos deviations of such amount that one image from each source 
is brought into juxtaposition. Since the holes a, b are at the focus 
of the lens L 3 the images produced by md lens are also circular 

but the biprism • into semicircles. Six out of the ei 

images are stopped out while the remaining two are observed through 

If the two beams are of equal intensity a uniforj nainated 

disc with a diametrical line is observed if the plane of polarisation of 
nicolpri : i 3 with the plane oE polarisai 

onent. Rota icol in either direction diminish^ 

: ;htness or one image and augments that of the other. If the 
ants are of unequal intensity n: [s affected by i i 

the nicol prism in either direction. A «a attac 

to the analyser ' ■ rve the angle 

Let the angle at which radiation from any source matches the 
lump be <#>. Then it: can be shown that 

In tan #-=fl-f--= . . • * ■ (IS) 

where 7" is the absolute temperature and a 3 b are constants. If we 
determine two values of $ corresponding to two values of T. we get 
3 straight line from which the temperature for any value of $ can be 
d. In practice, the pyrometer is calibrated by a direct comparison 
ith a standard thermometer, and the disc is directly graduated in 
deg;" 1 

RADIATION FROM THE SUN 

38. The Solar Constant.— The sun emits radiant energy conti- 
nuously in space of which an insignificant part reach: rth. But 
even of this incoming radiation, a considerable portion is lost by 
.: and scattering by the terrestrial atmosphere and is sent back 
i the interstellar space/ The best reflecting constituents of 
atmosphere are water, snow and cloud. The scattering- is partly due 
to the dust particles and partly due to the air molecules and is gener- 
,' small. Further the radiation is heavily absorbed by the earth's 
■sphere, the total absorption varying from 20 to 40% depending 






RADIATION 



[CHAP. 



the day and the season of the year, and the different 
parts of die spectrum are absorbed to a different degree. We 
:. Eor a more constant quantity which is furnished by 
h solar radiations are received by one sq, cm, of 
surface held at right angles to the sun's rays arid placx 
the mean distance of the earth provided there were no absorptio 
the atmosphere or provided the atmosphere were not present. This 
is call . ant and is generally exp in calories 

minute i shall now give a method 'of determining the 

constant, 

39, Determination of Solar Constant. — The Absolute Pyrheliometer. 

— Among the early workers who attempted > neaaure the 
constant were Wilson. The instrument; 

which the solar radiations arc measured are called pyrheliometers. 
We shall describe the mater-stir . . smpioyeti in the Astro- 

B 




Fig. 25. — The wa1 iter. 

physical Observatory of the Smithsonian Institution, Washington, 
a cross-sectional view of the apparatus. 
ilacfc-body chamber for the reception of solar radiation, whii 
further protected from air currents by a vestibule, not shown. This 
is simply a hollow cyli need in front of AA. The chamber is 

blackened inside and has its rear end of conical shape, ai 
surrounded by water contained in the calorimeter DD which is 51 
nisly by means of a stirring arrangement BB run by an el 
motor from outside. C is a diaphragm of known aperture £< 
ing the solar radiation. The incident radiation is completely absorbed 
by the chamber producing a rise in temperature of the water contained 
in the calorimeter. This rise is measured by the platinum resistance 
thermometer F whose wire is carefully wound upon an insulating 
frame round AA; At E is inserted a' mercury thermometer. The 



XI.] 



DKIT-KMLXATION Ql' SOLAR CONSTANT 



SIS 



calorimeter DD is carefully insulated from thermal effects occurring 
■ 1 : 

For calibrating the instrument, a known amount of electrical 
energy is supplied to the manganin resistance wire G and the rise in 
tempera t tire noted. Thu eat obtained by solar radiation can 

compared with the heat generated electrically* 

Another type of pyrheliometer is called the water- flow , 
meter. these a steady stream of water hows past the absorption 

chamber and die temperatur 'ween the incoming and 

water is observed. 

Instruments are called absolu 
they measure the enerj v. For most purposes it is more con- 

ient to employ secondary pyrheliometers in which radiation is 
;rbed by a blackened silver disc. In the 'compensation pyrhelio- 
meter of Angstrom there are two thin strips, of metal idem:. I : 
every way which serve as the calorimetric body. One of these is 
sosed to the sun while through the other, which .is shielded by a 
screen, an electric current is passed. The strength of the current is 
regulated that the temperatures of the two strips, as indicated by 
thermocouples attached in opposition, is the same. The energy of 
incident radiation Is then equal to the electrical energy suppli 
Wing the length and breadth of the strips and their ion 

incident radiation per unit area can be calcula 
ith the help, of these instruments we are able to measure the 
rati: reived per minute at the earth's surface and from this 

w ? Iculate the solar constant The radiation received vari 

with the time oF the day, depending upon the zenith distance of the 
sun*| If we assume the absorption to be due to an atmosphere 
homogeneous composition, then applying- Biot's law, we have 

7 = I, (TV, [l\ 

where d is the thickness of the medium traversed, I (> , I the intensity 
of (hi beam just at the beginning and the end of the medium. This 
d strictly for a monochromatic ray and a homogeneous medium. 
In t. e atmosphere there is no homogeneity in dust content or i 
densiy in a vertical direction. In the actual' e: periment the intensity 
of sdar radiation as received on the earth is oh liferent 

elections of the sun on the same day with constant sky conditions. 
rhei d varies as sec z where z h the sun's zenith distj 1 urther 

ling as a first approximation* that k t is the sara . ! wave 
hs we can write 

S - S,, a«« ■, (20) 

v., S represent the true and the observed solar constant'res- 
[y. and a is called the transmission coefficient and varies from 
0.551 to 0.85. Then taking logarithm* 



assu 

lend 



Then taking logarithms 

log; S — IogS -psecz log a. 






I- or •:•.:• ate work h or a i, not assume 1 
•o -.| w 8 **? 08 ™ se Parately and from this extrapolated curve the vali 



RADIATION 



[CHAP, 



Plotting fhe values of loo- S as ordinate* and the corresponding values 

of ser z as abscissa we obtain a straight line wjjose intercept on tne 

ate axis gives log 5,> whence £<i is found, 

. curate experiments give the mean value c \ the i snstant to he 

1.937 i - per minute per sq, cm. The i int i found to 

.en-year cycle of the sun. 

40. Temperature of the Sun.— The sun consists of a central hot 
linating in a surface called the 

sion 'temperature of the sun' we generally mean the temperature of 

-c The temperature inside the central core is, how 
milc h ! n this. We shall now describe some sin 

for de ig the temperature of the photosphere of the sun, 

on measurement of radiation. 

41. Tamperatwe from Total Radiafiwu— According to cat 
serrations by Abbot and others the so 

mte per cm 2 . We can now find out what wouli ■ temjera- 

nrface free from an sre and havir 

s \ om ■ v.- which would emit the same tota 

ceof the earth, This naturally 
for the ture of the photosphere. 

Lei the radius of the sun be r, a a '- of heat loft by 

|" 
H — '\ir^<iT\. 

u. If we Lejcrjbe 

is R concentric with the sun (R being the d 
ill be spri ad over 
d per unit surface of the earth is 






4irr 2 

4'jtK 2 



*T*, 






-=(f)v 



(22) 



Now r/R, the r an i ngulaT radius of the sun — 959" = 4.49 >( 
fans ; cr— 5,77 y 1 fr 5 ergs sec.- 1 cm." 2 degree-*: S = 1-957 cal. 
cm. -3 min."" 1 . Hence we obtain after substituting these vahss in 
that T — ain2 c }L. 

42. Temperature from Wavelength of Maximum Emission.—- 
Wien's displacement law A M T = 0.2884 cm, X degree (eq i 14) 

also be employed to determine the temperature of the 
sun. Abbot's investigations show that A m z=475S A, U., wience 
T = G059--K. This temperature Is about 300° higher than flu tem- 
:• lire deduced from total radiation. Thii i gence is iasiiy 
understood if we remember the fact that contrary to il u otion 

made here, the sun docs not actually radiate like a blai I .iy. 

43. Temperature of Stars. — The stars are so many suns, only 
they are at enormous distances compared to the sun. The detemina- 



XL\ 



TEMPERATURE OF STARS 






tion of their temperature is subject to the same uncertainties as in the 
case of the i Eollowed are. almost identical ex 

that her :otal radiation method fails as it is often no 

line the diameter of stars. The temperatures are :' 

intensity in their emission s] pplica- 

tion of equation (14), but the actual methods are far more complii 
The temp u ' : a|ltl othei ways - 

from 2501)'" C (red stars} to nearly 30,000° C (for bluish white st 

Books Recommended* 

i, GlazebrooTc, A Did' ,' Applied Physics, Vol. 4, Arti 

by Coblentz. 

2. Planch, WSrmestmhlung or English translation by Mash 

;). E. Griffiths, Methods of . • Temperature. 

■f. Bur id Le Ch: telii , T " isurement of High 1 

ure. 

Abbot, The Sun. 



CHAPTER XH 



THERMODYNAMICS? OF THE ATMOSPHERE 

1. ] ii the present chap; tiall consider the application of the 
principles already developed in this book to the earth's atmosphere. 

e doing so, however, we shall first state .some of the results 
already known to us from the study of the atmosphere. 

DU ution of temperature 

2. Vertical Distribution of Temperature. — The results of aerolo- 
observations show that the temperature of the atmosphere 

as we go up in the atmosphere. By combining 
several observations made over a given region, the form of the curve 



:i»2[C JI2208 Jos 



20 




Temperature*/* 

1 UJvJuqo F ff-. c :cmpevelaT« 



taia =10*A 



1 i- — Vertical distribution of monthly mean temperatures ever A 

g the variation of temperature with altitude is determined. This 
curve varies slightly with the season, particularly at the lower levels. 
the vertical distribution of the monthly mean temperature 
over Agi a (lat. 17° TO'N, long. 78° 5'E) in the Uttar Pradesh of l 

The fall of temperature due to a rise of 100 metres is usually 

called the vertical gradient of temperature and the fall of temperature 
Homed* rise in altitude is usually known as the lapse-mlz of 
temperature. The lapse-rate over any particular region varies with 
altitude and there is also a seasonal variation. 

3. Troposphere and Stratosphere. — From Fig. 1 it is evident that 

Agra up to a height of 13 to 16 km. there is a rapid fall of 
rature, above which., however, the rate of fall dirnxni 
at a height of about 20 km. the temperature becomes almost constant. 



xii,] 



TROPOSPHERE AND STRATOSPHERE 



317 



i.€. t the lapse-rate vanishes. A similar discontinuity in the vertical 
distribution of temperature in the atmosphere is noticed all over the 
■ :j.Id, although the height at which it occurs is not the same ei 
where. The outer shell of the atmosphere, in which the tempers 
remains practically constant with vari i of height, is. gi'mi 

ecial name ol •' foe Zone or Is 

to distinguish it from the Tr> re or the Convecth which 

is the lower portion in. which there is a considerable fall of tempera- 
ture with height. The surface o of these two region'- i 
the atmosphere, which plays a very important role in modern 

atmospheric circulation, is railed the tropopause. TV i he of 
the tropopause varies with the latitude. The Iropopau ■ is to 

lower towards the ground as we proceed from the equator to the 
poles, its heighJ being about M km. ii the equator and about 8 to 10 
km. at the poles and is greater in summer than in winl 

: causes of the diminution of temperature with height in the 

troposphere are manifold. Let us try to explain here in general terms 
why this temperature diminution occurs. The solar radiation 1 
its pas^ge through the earth's atmosphere Is only slightly al 
It, and the amount a isorbed d over such a larj 

air that the latter is not ■■■ heated by i 

radiation. In contrast to this, however, the energy received by 
earth is concentrated and therefore heats its surface considerably. 
ie heated surface in turn warms the air above it, parti) by contact 
and partly by the long wavelength radiated by it and absorbi •' '•■ the 
air. Now the temperature of afr at any height depends upon the 'total 
energy absorbed and emitted by it. The tower a ton | :i ^ sri 
temperatures emits more energy than it a and therefoi I 

toco I :■ radiation. These two phenomena, the h iti of the earth 
and the cooling of the layer above, so affect the density of the atmos- 
phere as to cause vertical convection, in consequence of which the 
warm ascending air becomes cooled through adiabatic expansion ami 
the descending air becomes heated by adiabatic com ires ion since 
the pressure or the atmosphere decreases with elevation. In tin's v 
the decrease of temperature with increase of elevation is established 
and maintained throughout the region in which vertical Jon 

takes place. We shall calculate in § 5 an expression for the lapse-rate. 
Above n certain height, however, convection becomes feeble ami 
the temperature of die atmosphere falls so touch that the heat radiated 
by it becomes equal to the amount absorbed by It either directly from 
the earth's radiation or from The p 'radiation. (This is be- 

cause the heat radiation received by the air from the earth reman 
practically constant at al] available altitudes) . The temperature of the 
layer remains practically unaltered at and above this b and 

therefore the convection currents cease above this heij I he 

stratosphere therefore is a result of the cessation of convection c 
rents and of establishment of radiative equilibrium. 

*This has already been discussed on pp. 311-313. 



318 



[IMODYNAWICS OF THE ATMOSPHERE 



! 



4, Vertical Distribution of Pressure.— The theory of the 

tton of pressure with increase of altitude is based on the applies 

of the gas laws Ei losphere. Lei: ua assume thai 

:re is in convei juilibrlum while the - 

e is in isothermal or ra equilibrium. Od account of 

convection currents the composition of *he air in the troposph 

ctically the same at ail I or the tl ical calcula; 

tion of pressure we can assume a mean gas constant for the atmos- 
pheric air- For the stratosphere hou \\ mvection currents cannot 
be assumed to exist since wa have assumed the layer to be isc 
hence no fixed value can be assumed for the gas constant R/M. We 
must therefore treat each constituent separately with its proper gas 
constant. 

Let -dp be the decrease of pressure corresponding to an increase 
vation dz. Then 'if p is the density of air at the point under 
consideration we have, on equating the decrease of pressure to che 
tit of die column dz f 

-dp^zgpdz. . ■ (1) 

From Boyle's and Charles' law we have h — pRT/M where M U die 
■ [ght of the gas. Hence (1) yields 

(- ; ) 






RT 






m r 
) p -' r\t * ■ ■ 

T to be constant, which is far from being the case as far 
posphere t on integrating i result 

P = pc r (•!) 

This is known as irmula and gives the pressure p at the 

height z in terms of the pressure /* ft at the earth's surra* 

As the temperature of the air column is really not constant, in 
actual practice the mean temperature, of the air column is substituted 
in equation (4) , This equation can then be readily applied to cal- 
culate the difference in height between two stations when the baro- 
metric pressures at the two stations are known. 

For calculating the distribution of pressure in the stratosphere 
we have to apply equation (4) to each constituent scp 
due to variation in composition M does not remain the same at all 
is. We therefore obtain ibr the partial pressures the result 

p = p t e-'W IT ;p=Po-e-* >i '< ! * T . ■ - (5) 

and the total pressure is the sum of these partial pressures, 

5. Con vective or Adialwitic Equilibrium.— We have already 
in S 3 that the troposphere is mainly in convective equilibrium. Hence 



XI!.] 



WATER VAPOUR IN THE ATMOSPHERE 



319 



■ ion (4) which was deduced on the assumption that the ;. 
phere I r . at rest ia not quite correct. The fact h really ■ : 

mines the distribution of the atmosphf isation oJ 

die condition tha 
on being moved from cue place to another, shall take up the 

sure in its new position without, any loss or gain of 
. taction. Tin: law • ting he volume and pre 
in the hould on such assumptioj :imate to the 

atic law. 
If the adiabatic law p = cp* holds for the atmosphere, we 

da 



dp A dp 

iz 



Hence from (1) we get 



cyp- 



-i* - 



Integrating- this v 



-: -<*-!)»,* 



(6) 



where p () is the de at zero level. This is the law according to 

which the density shoi id i 11 off with increase in height in the tropo- 
sphere. Since >RT/M, we get on substitution in the relation 
p = cfP the result 



T = 






R 



-or-1 



R(Tr-T) 

■ - ess 



titu ting this in (6) we obtain 

■ T is the temperature at height zero, and J llie mechanical 
equi , alent of Heat. Now since y = c p je v wd JM {n p -r v ) — R, the 

: relation yields* 

... (V) 






Jr 



10293' 



Thus the temperature decreases proportionately to the increase of 
height as we yo upward;:; in th< re. Substituting numeric ■' 

■ find that the constant of the above equation is about 
a. This value is about twice the experimentally observed tem- 
per;!; tamely 5°C. per km. 

WATER VAPOUR IN THE ATMOSPHERE 

6. Hygrometric State of air. — Water gets into the atrai 
:i account of evaporation from surfaces of oceans, lakes, 

could be (25), p. -: : 

I :-i::;:lii n (23), 






S or THE vi -• 



CHAP. 



ered mountains, moist soil and, from various other sources. 
iration d pends upon a number of factor:-.;, na e 

rature of the air, eloci y, the pressure and the amount of 

i ' < i : litis. Increase of temperature ai 
wind velocity increase aporation while increase of pres- 

sure and of moisture in the atn decreases it. But the 

;ity of the air to I, r vapour fa limited and depends upon 

temperature only, At a temperature i, air can hold only a o 

: pour which is giv< i the satun :ipour 

ure corresponding to the temperature t, and this anu it 

• of temperature. In Table i w ..i-ated 

vapour pressure of water at different temperatures and the values are 

Table 2. — Maximum vapour pressure of water in mill-'. 
of ' ent temperatures. 



D C 


o 


1 


2 


3 


4 


5 


6 


7 


8 


y 





4,579 




5.2M 




6.101 : . 


7.0^ ?.SI3 8.045 




10 


9.21 






11.23 


• 


. 










. 


19.8 


23.76 












, 






37.73 


39.90 42. IS 




■•v.-'/ 


49.69 








2 


•1 








12 


79.60 




#8.02 







6 


8 


10 


14 


16 




30 






1 3.8 


149.4 


163,8 


179.3 


196,1 


214.2 








A 327..1 355.1 




S 450.9 


■:\ 






. 


610.9 6^7.6 7C7.3 760.0 




875.1 937.9 1004,4 



plotted in Fig. 2, p. 321. If the air contains the maximum amount of 

water vapour that it can hold, it is saturated ; if it contains a k 
amount, it is unsaturated. In some cases, it may contain more than 
the equilibrium quantity; it is then called supersaturated. The 

eld in air may be expressed in gra a : c 
cubic metre or in tern essure in millimetres which it exerts. This 

is known as the "Absoluts Hum ic air. Humidity thus con, 

information regarding quantity of water vapour in the atmos 
The state of the ihere with regard to its actus ttent 

is generally expressed by its Relative Humidity, which is tin 

ctual quantity i pour present in a divert quantity of 

air to the maximum quantity that it could hold if it were sal 

'he observed temperature. Relative K 

in • the hygrometric state of air Le., its moisture 

itent and hold moisture ran be fully specified by 

ire and relative humidity. 

7. Dew-poir.. — If air containing moisture is proj cooled, 

a temperature wilt be reached at which the moisture that it contains 






X1L] 



HYGROMETER 



321 




IQ 20 SG 40 50 60 10 W Wlw-C 

Fig-- 2- — Vapour pressure euirve o£ water. 



is sufficient to saturate it. This temperature is called the Dew-point. 
Any further cooling of the air will bring about a deposition oE moisture 
on the surface of the containing vessel in the form of "dew". In 
the large-scale phenomena 
occurring in the atmosphere the 
deposition may take any one of 
the different 'forms, viz., Bog, 
cloud, rain, frost., hail, snow, 
dew, etc:. It is easily seen that 
the four quantities — tempera- 
ture, absolute humidity, relative 
humidity and dew-point are 
inter-related and a determina- 
tiorj of only two of them is suffi- 
cient. 

This can be easily done 
with the help of Table 1. Thus 
if the temperature and the dew 
point are 30" and 20°C. res 
tively, the saturated vapour 

iure at 20 Q C, is 17.54 nun. 
which gives the absolute humi- 
dity of the air, and the re] tive humidity (tt.H.) is (17,54/31.82) X 
100 — 55%. Conversely., if the absolute 'humidity is known to be 9.7 
mm. the dew-point is seen from the table to be equal to 10°C, and 
the R.H.= (9.7/3U2) ;< 100 = 30%. 

8, Hygrometers. — The study and measurement of moisture pre- 
sent in the atmosphere is called Hygromeiry and the instruments 
used for measuring the amount of moisture are called hygrometers 
[Hys?o — moisture, meter = measurer) . From what has been said 
above, it will be evident that besides temperature we need measure 
any one of the three quantities — absolute humidity, relative humidity 
and dew-point. This gives rise to a variety of hygrometers which may- 
be broadly classified as follows:— (1) Absorption hygrometers such 
as the chemical hygrometer, (2) Dew-point hygrometers, (3) Empiri- 
cal hygrometers such as the wet-and-dry bulb hygrometer and the 
hair hygrometer for which no complete theory has yet been worked 
out but which are by far the most important. 

9. The Cbemical Hygrometer, — Absolute humidity is directly 

found out by means of the chemical hygrometer. In this instrument 
a stream of air is aspirated slowly over drying tubes, and the gain 
in weight of the tubes and the volume of air passed over are recorded. 
Thus the mass of water vapour actually present in a given volume of 
air is found out, and this is compared with the mass required 10 
saturate the same volume of air at the same temperature. Tin's biter 
quantity is given in tables. 

The disadvantages of this medio apparatu 

21 



S22 



THERMODYNAMICS OF THE ATMOSPHERE 




CBAP. 




and not easily portable, the experiment takes considerable time and 
laborious corrections must be applied. Its chief use is for standardis- 
ing the simpler types of instruments in the laboratory. 

ID. Dew-point Hygrometers.— Hygrometers in which humidity 
is found from a direct determination of the dew-point are called Dew- 
point Hydrometers. Examples of this type are the Daniel!, the 
Regnault'and the Dines* hygrometers. The essential principle under- 
lying all ol them is the same, viz., a surf ace exposed to air is steadily 
cooled till moisture in the form of dew begins to deposit on it. The 
temperature is again allowed to rise till the dew disappears. The 
mean of the two temperatures at which the dew appears and dis- 
appears gives the dew-point. These hygrometers are however rarely 
used in meteorological work. They differ from one another in the 
manner of cooling or in the nature of the exposed surface. We shall 
therefore describe only one of them, viz., the Regnault's hygrometer. 

11. Regnault's Dew-point Hygrometer.— This consists of a glass 
tube fitted with a thin polished silver thimble S (Fig, 3) containing 
ether. The mouth of the tube is closed by a cork through which passes 
a long tube going to the bottom of the ether, a thermometer with its 
bulb dipping in the ether, and a short tube T connected on the outside 

to an aspirator. When the aspira- 
tor is in action air is continuously 
drawn through the ether producing 
a cooling and the temperature of 
the thermometer falls. The pto- 
cess is continued till moisture 
deposits on the surface of the 
thimble, and the corresponding 
temperature is noted. In order to 
help in recognizing the first ap- 
ance of this moisture by com- 
parison, a second similar tube pro- 
wl th a .silver thimble S but 
without ether is placed beside it. 
Next the aspirator is stopped, the 
apparatus allowed to heat up and 
the temperature when the dew 
disappears is noted. The mean of 
these two temperatures gives the 
dew-point. The dew-point hygro- 
meter has the following disadvan- 
tages ;— (1) It is difficult to deter- 
mine the instant when dew appears, (2) the temperature of the ther- 
mometer does not accurately represent the temperature of the surface, 
(B) the instrument should be used in still air, (4) the observer him- 
self, being a source of water vapour, is likely to disturb the readings. 
Attempts have been made to minimise or eliminate these difficulties, 
but it is not possible to get continuous records from such instruments. 




Fiff. 3.— Regnault's 
Dew-point Hygrometer. 



XII.] 



WET AND DRY BULB HYGROMETER 



S23 



12. The Wet and Dry Bulb Hygrometer or P*yehromet«r.— 
"Relative Humidity" can be easily measured by means of a wet and 
dry bulb hygrometer. This consists of two accurate mercury thermo- 
meters suitably mounted on a frame. Round the bulb of one of these 
is tied a piece of muslin to which is attached a w T ick extending down 
into a vessel containing pure water. The evaporation from the large 
surface exposed by the muslin produces a cooling and thus the wet 
bulb thermometer records a lower temperature than the dry bulb 
thermometer. In the steady state there is a thermal balance between 
the wet bulb and the surroundings. The greater the evaporation the 
greater will be the difference in temperature between the two. Now 
evaporation will be greater the lesser the humidity of the air and thus 
the difference in temperature between the wet bulb and the dry bulb 
is a direct measure of the humidity. The rate of evaporation is, how- 
ever, further affected by the pressure and the wind; large pressure 
tends to retard evaporation while large wind velocity accelerates it 
The effect of pressure is however very small and may be neglected, 
while the effect of wind is rendered constant by maintaining a constant 
supply of fresh air. 

A relation between the readings of the two thermometers and 
certain other quantities can be easily found. If T, T' denote the 
absolute temperatures of the dry and wet bulbs respectively, p the 
pressure of water vapour prevailing in the air and p* the saturated 
vapour pressure at T' s and 11 the barometric pressure, the rate of eva- 



poration will be proportional to 



£-p 
H 



and also to (T - T) ; therefore 



p'-p = AH(T-T'), 



(8) 



where A is some constant depending upon the conditions of ventilation 
and is determined from, a large number of experiments. In actual 
practice Ilygrometric Tables have been prepared by the Meteorological 
Office assuming the value of A for a fixed draught of air. From these 
tables the pressure p of water vapour prevailing in the air can be 
directly read if the dry bulb temperature and the difference between 
tlie dry and wet bulb temperatures are known. Knowing p the rela- 
tive humidity can be found, 

13, The Hair Hygrometer. — For ordinary purposes the relative 
humidity can be roughly measured by the Hair Hygrometer. This 
consists essentially of a long human hair from which all oily substance 
has been extracted by soaking it in alcohol or a weak alkali solution 
(NaOH or KOII) . When so treated die hair acquires the property of 
absorbing moisture from the air on being exposed to it and thereby 
changing in length. Experiments have shown that this change in 
length is approximately proportional to the change, between certain 
limits, in the relative humidity of the atmosphere. Fig, 4 shows a 







THERMODYNAMICS OF THE ATMOSPHERE 



CHAP. 



hair hygrometer. The hair h has its one end rigidly fixed at A while 
the other end passes over a cylinder and is kept taut by a weight or 

spring. The cylinder carries a pointer 
whidi moves over a scale of relative 
humidity graduated from to 100, 
The changes in length or the hair due 
to changes in humidity tend to rotate 
the cylinder and thereby causi a 
motion of the pointer. The instru- 
ment must be frequejjtiy standardiz- 
ed by comparison with an accurate 
hygrometer and then its readings are 
reliable to within 5%. 





Kg. 4. — Tl meter. 



14. Methods of Causing Con- 
densation. — We shall now find out 
under whaf conditions the water 
vapour present in the atmosphere 
be precipitated from it. This water 
vapour can be condensed into liquid 
water or solid ice if the actual vapour 
exceeds the maximum vapour 
pressure corresponding to the exist- 
ing temperature, This happens al- 
i exclusively when the air is 
cooled down more or less suddenly 
but in rare cases it may occur if the 
jure happens to increase 
■lIill: to some local h as 

compression of saturated wate etc. 

The cooling of air may take place by the following three 
processes : — 

(1) Due to radiation of heat or due to contact with cold bodies. 

(2) Due to the mixing of cold and warm air masses, 

(3) Due to adiabatic expansion caused by sudden decrease of 
pressure. 

The first process should have been the most effective in produc- 
ing precipitation had it been active in large masses of air. But air 
even when it is moist, is a poor conductor and radiator of heat, so 
that radiation and conduction of heat play a minor roTe in the pheno- 
menon of precipitation. The result o£ the loss of heat by radiation 
or by contact, with cold bodies, such as the surface of the earth in 
winter, cold walls, stones, etc, is the formation of mist, fog, dew, etc. 

The second mode of condensation depends essentially on the 
experimental fact that the saturated vapour pressure of water increases 
much more rapidly with increase of temperature than the temperature 
itself. Thus if two equal masses of air, initially saturated at tern- 






XII.] 



ADIABATIC CHANCE OF HU.MlI> AIR 



325 



peratures t and ¥ respectively, are allowed to mix together, they will 
acquire the mean temperature t m = (t -|- F)/2, while the mean vapour 
pressure will be (e + e*) /2, where e, e' denote the saturated vapour 
pressure of water at temperatures t and t' respectively. On account 
of the above property, however, this mean vapour pressure will be 
greater than E, the saturated vapour pressure at t m and therefore 
the excess of water will condense. As an example take the following 
illustration : — Let us have equal masses of saturated air at 4° and 
82°C. When mixed up the temperature becomes 18°C. To saturate 
the mass we require 15.4 gm. per cubic metre. The separate masses 

contain 6.4 and 33,8 grams and the mixture ,/' - gm. per cubic 

metre. Hence 20.1 — 15,4 = 4.7 gm. will separate by condensation. 
If the two masses of air are not saturated before mixing, there 

may be condensation in some cases. This will depend upon the 
proportions of the mixture. If both the masses are very near the 
point of saturation, then condensation may take place at some places, 
and no condensation or even evaporation at others. This explains 
the formation and disappearance of certain kind of clouds. 

This third process is the most important because it is active on 
a large scale and produces cloud and rain. When moist air is 
allowed to expand adiabatically its temperature falls and some of 
its moisture is condensed if the temperature falls below the dew- 
point. This is the process which generally takes place in the atmos- 
phere. An ascending current of moist air suffers a decrease of 
pressure as it ascends ; it therefore expands almost adiabatically and 
partis with some of its moisture. To calculate the cooling we have 
mass of air, 

IS, Adiabatic Change of Humid Air,— From the first law of 
thermodynamics if dQ be the amount of heat supplied to a given 
mass of air, 



«-*JT--}*-**T- 



(9) 



In case of a mass of saturated air rising upwards, the heat dQ 
is added as a result of an amount dm of vapour being condensed. 
Hence, 

d(l= -Ldm, (10) 

where L is the latent heat of vaporization. Therefore 



—Ldm = CpdT — '' ' 
MpJ 



(II) 



The total mass m of water vapour in the air per c.c« is given by 

m = 0*623— xp, 

P 



THERMODYNAMICS OF TILE ATMOSPHERE 



[chap- 



where 0.623 is the ratio of the molecular weight of water vapour to 
the weighted mean of the molecular weight of the constituents of dry 
air, e the vapour pressure, and p the pressure of the dry air and p 
its density. Hence 

dm de dp 

m e p 

Substituting this value of dm in (11), we have 

de- I.m . RT 

-Lm T + - dp - H dT+ j^jdp - 0, 



(12) 



or, 



Now dp=i -pgdz = —jppdz* Substituting this value in (13) 



we get 



dT = _ s \ RT ^ Jf 

dz , Lm de 

e >+-TdT 



(14) 



This is the rate of decrease of temperature with elevation of 
saturated air. All the quantities on the right-hand side of this 

equation are known, so that — — can be easily evaluated. 

Books Recommended. 

1. Humphreys, Physics of Air. 

2. Brunt, Meteorology. 

3. Lempfert, Meteorology, 

4i Hann, Lehrhuch der Meteorologie. 

5. Wegener, Thermodynamik der Atmosphere. 






APPENDIX 1 

ERRORS OF MERCURY THERMOMETER AND 
THEIR CORRECTION 

As mentioned on p. 2 various corrections must be applied to 
the mercury thermometer if it is used for accurate work. The method 
of applying these corrections is explained below : — 

(i) Secular Rise of Zero. Glass is to some extent plastic and 
therefore its recovery to its original volume is an extremely slow pro- 
During the construction of the thermometer the glass is heated 
to high temperatures and then allowed to cool. In this cooling pro- 
cess the contraction of the glass first takes place rapidly and then 
slowly even upto several years. Naturally therefore when calibration 
of the thermometer is usually undertaken the glass has not contracted 
to its final steady volume and the zero-point shows a secular rise for 
years due to this gradual contraction. This defect can be greatly 
removed by choosing suitable material for the glass of the thermo- 
meter, by properly annealing the tubes and storing them for years 
before making thermometers out of them. 

(it) Depression of Zero. This defect is also due to the defect in 
the property of glass mentioned above. When a thermometer is 
suddenly cooled from 100° to 0°C, the bulb does not at once regain 
its original volume and there is a consequent depression of zero, - 
magnitude is greater the higher the temperature to which the ther- 
mometer was exposed and the longer the duration of this exposure. 
The method adopted by the Bureau International to correct for the 
depression of zero is the "movable-zero method" of reading temper a- 
tures. In this method the boiling point (100°C) is first determined 
and immediately after, the ice reading is taken ; let these readings on 
the thermometer be X and Z respectively. Suppose this thermometer 
reads X s when immersed in a bath at (°C. Immediately after this 
the thermometer is immersed in ice ; let its corresponding reading 
be Z ( , Then the correct temperature f°C of the bath is given by 



x-Z 



100, 



(Hi ) Errors in the fixed points. For the lower fixed point the 
thermometer is clamped vertically with the bulb and a little part 
of the stem surrounded by pure ice mixed with a little quantity of 
distilled water. Suppose in the steady state the mercury stand's at 
-0.PG ; then the freezing point correction Is -j-0J o C (additive). If 
the level of mercury stands above the zero degree mark, the correc- 
tion is subtractive. 

For the upper fixed point the thermometer is kept suspended 
inside a hypsometer with the bulb exposed to steam in the inner 
chamber. The steady reading of the mercury level is observed and 
the reading of the manometer indicating the pressure of steam noted. 
Suppose the thermometer reads no .2°C"under a pressure of 75.8 cm 



328 




APPENDIX 



of mercury. Since the boiling point for this pressure is 99.93°C, the 
correction is 99.93 -99.2 = 0.73°G and is positive. If the obse 
boiling point i.s above the calculated one, the correction is negative. 
(jV) Correction fornon-itnift ire. As capillary tubes are 

drawn and not bored, slight inequalities in the diameter of the bore 
are bound to exist in the atem of the thermometer. This necessi atea 
a small correction which is carried out as follows : — A small portion 
of the mercury thread is detached from the rest and its Le 
measured when it occupies successively different parts of the stem, 
say between and 10, 10 and 20, ♦ . , 90 and 100 marks. The 
measured lengths will vary from place to place due to non-uniformity 

of the bore ; let these lengths be t it l 2 lw> respectively. Let the 

corrections to be made for non-uniformity in the vicinity of the 

0, 10, 100 mark be a^, «ki «iocj respectively. If I is the accurate 

length of the mercury thread used, 

J = / x -t- a 10 - % . . . . (1) 

J = 'a + *ao-»io ■ ■ • • ( 2 ) 



I = JiO + «1W» - «B0 ■ • • (3) 

Adding up, 

10 I = & -f- J s + r 10 ) H- C^joo - flo) • • • (4) 

a 1OT and a n are the corrections to the upper and lower fixed points 
which can Trained experimental!) as explained in (Hi) above. 

Hence / can be calculated from ('!) . Substituting this value of / in (1) 
«,o can be calculated since a (t is known. Similarly from (2) a 

then di awn with the marked 
divisions i . a vil ..... a vl[ as ordinate. From this 

-.i Lures can be easily rend, 

In li thermometers this tedious correction has 

not to be applied by the user as the interval between the fixed points 
is subdivided not into equal parts but into equal volumes to represent 
the degrees on tins thermometer. 

(v) Correction for lag of the thermometer. If the bulb of a 
thermometer is placed in a hot bath, the thermometer will not attain 
the temperature of the bath instantaneously will require a small, 
definite interval of time to attain that temperature. This is called 
the "lag" i I the thermometer. The lag of the mercury thermometer 
increases with the mass of mercury and the thickness of the glass 
and also depends upon the nature of the medium surrounding the 
bulb. Due to this lag the thermometer reading will be higher when 
a bath is cooling and lower when the bath temperature is rising. 

Suppose we consider the case when the temperature of the bath 
is rising. The correction for lag is applied as follows: — The bulb of 
the thermometer is immersed in the bath and the thermometer read- 
ings noted as a function of time and the observations plotted on a 
graph with temperature as ordinate and time as abscissa (Fig. 1) . 
The curve AB represents the rise of temperature with time. To 



ERRORS OF MKKCLRY THERMOMETER 



: 'J 



find the correct temperature of the bath at each point on the curve, 
an auxiliary experiment is performed in which the bulb of the thermo- 
meter is immersed in a thermostatic bath maintained at a temperature 
somewhat higher than the maximum recorded in the main experiment. 
The thermometer readings are read at short intervals until the thermo- 




Tlrne 



A" ^ 



Time 

Fig. 1 




Fig. 2 



meter attains the steady temperature of the bath. These readings are 
plotted on a graph with time as abscissa and the difference of the 
h; Miometer reading and the temperature of the bath as ordinate, 
and the curve RS (Fig, 2) obtained. 

I l will now be assumed that the lag does not depend upon the 
actual temperature of the bath but only upon the rate at which 
the temperature is changing. Consider a point M on the main 
curve. The lag at this instant depends upon the slope (dO/dt) of the 
curve at this point M. Now find out a point P on the curve RS where 
the slope is the nine as the slope at the point M, Then the lag PQ 
will also be the lag at M. Hence if PQ s= MN, N gives the correct 
temperature corresponding to M. In this way the corrected curve 
A'B* can be easily drawn. 

(vi) Error due to changes in the size of the bulb caused by 
variable internal and external pressure. For diminishing the time 
lag the bulb of the thermometer is usually made thin, Am increase 
in external pressure therefore easily alters the volume of the bulb 
and causes a rise of mercury level in the stem. Suppose the thermo- 
raduated when the external pressure is equal to the atmos- 
pheric pressure, If the external pressure is now* increased the bulb 
will contract and the mercury En the stem will rise. The external 
pressure coefficient is defined as the ratio of the rise of mercury in the 
expressed in degrees, to the increase in external pressure, 
expressed in mm. of mercury. This ratio can be easily determined 
experimentally. Knowing this and the external pressure to which the 
bulb is subjected at the time of reading the thermometer, the correc- 
tion to be applied can be readily calculated. 

When a thermometer has been graduated in the horizontal posi- 
tion and a subsequent reading is taken with the thermometer in the 
vertical position, there is an increase in the internal pressure due to 
the vertical column of mercury in the thermometer. The bulb con- 



330 



APPENDIX 



sequently expands causing a depression of mercury in the stem, for 
which a correction is necessary. To determine this correction the 
readings of the thermometer are observed in the horizontal and vertical 
positions at any one temperature. The difference in the two readings 
gives the depression of mercury in the stem due to an increase in 
pressure caused by the mercury column which ex Lends from the centre 
of the bulb to the mark on the thermometer at which mercury stood 
in the vertical position of the thermometer. Expressing this pressure 
in mm. of mercury we can define the internal pressure coefficient as 
the ratio of the depression of mercury in the stem, expressed in 
•rces, to this increase of internal pressure. Knowing this coeffi- 
cient, which is constant, the correction can be calculated for any read- 
ing of the thermometer in the vertical position. 

(vii) Error due to capillarity. The surface tension of mercury 
causes an excess of pressure within trie meniscus over that outside. 
This excess pressure depends on the radius of the tube at the point 
where the meniscus lies {p cc 1/r). If the stem is not uniform in 
bore, there will be variations of internal pressure as the thread of 
mercury rises or falls and therefore the thermometer readings will not 
be very accurate. Further the angle of contact beween mercury and 
the sides of the tube depends upon whether the mercury is rising or 
falling, the meniscus being flatter when mercury is falling. Therefore 
a rising thread always gives somewhat lower readings than a falling 
one. it is also found that the mercury thread is less disturbed by 
capillarity when it rises than when it falls and therefore it. is preferable 
to take readings with a rising column. 

(viii) Error due to exposed stem or emergent column. Generally 
when the temperature, of a bath is measured, only the bulb of the 
thermometer and a portion of the stem are immersed in the bath. 
Tn such rnses the part of the stem exposed to the atmosphere does 
not acquire the temperature of the bath and therefore the thermometer 
reading will be less than the true temperature of the bath. The 
correction for this exposed or emergent column is applied as follows: — 
Let the thermometer reading he \ when the stem upto t 2 mark is 
immersed in the bulb. Thus n (■= f a — 1 2 ) divisions are exposed to 
the atmosphere, and its average temperature to is measured by a 
special integrating thermometer with a long bulb placed near it, with 
its centre coinciding with the centre of the exposed part. Let t 
denote the corrected temperature and B = m — g the coefficient of 
apparent expansion of mercury in glass (m = expansion coefficient of 
mercury, g, of glass) . Then a mercury column w T hose length is n 
divisions at * has to be corrected to the temperature t. If this 
column were to rise in temperature from t to *, the increase of height 
(which measures the increase of volume) would be «*{*—/<>) divisions. 
This must therefore be added to the observed reading t t to give t, 
Hence 



QT 



f a — n8t 
1—nB 



EXAMPLES* 
I 

1. Discuss the advantages of using one of the permanent g 

as a thermometric substance for defining a scale of temperature. 
Describe some convenient and accurate form of gas thermometer, 
explain its mode of use and show how the temperature is calculated 
from the observations made with it. (Madras, B.Sc.) 

2. The pressure of the air in a constant volume gas thermometer 
is 80.0 cm. and 109.4 cm. at 0° and 1()0°C, respectively. When the 
bulb is placed in some hot water the pressure is 94.7 cm. Calculate 
the temperature of the hot water, 

5, An air bubble rises from the bottom of a pond, where the 
temperature is 7°C,, to the surface 27 metres above, at which the 
temperature is 17°C Find the relative diameters of the bubble in 
the two positions, assuming that the pressure at the pond surface is 
equal to that of a column of mercury of density 13.6 gm. per cc. and 
76 cm. in height, 

4. Explain clearly the meaning of absolute temperature of the 
air thermometer scale. A gram of air is heated from 25 °C to 70°C. 
under a constant pressure of 75 cm. of mercury. Calculate the 
external work done in the expansion given that the density of air 
at N.T.P. is 0-001293. 

5. What is an air thermometer? Explain the method ol 
measuring temperature by Callendar's compensated air thermometer, 
Describe a method for measuring very high temperatures. (A. IT., 
B.Sc.) 

0. Describe the Callendar's compensated thermometer and 
explain how temperatures are taken with it. (A. LL. B.Sc, 1944 : 
Utkal Univ., 1952; Punjab Univ., 1954.) 

7. Describe briefly the method of standardisation and the range 
of usefulness of platinum resistance thermometers, and discuss some 
of the difficulties of precise resistance measurement and the precau- 
tions to be taken to avoid or correct for these, (A. U„ B.Sc. lions,, 
1929.) 

8. Give an account of the construction and use of the platinum 
resistance thermometer, pointing out any special advantages of the 
instrument. (Utkal Univ., 1950; Gujerat Univ.. 1951 ; Punjab Univ., 
1956.) 

9. Describe various methods of measuring high temperatures. 
(A. U., B.Sc, 1931, 1932, 1949; Gujerat Univ., '1951; Punjab Univ., 
1956.) 

10. State, with reasons, the type of temperature measuring 
device which you consider most suitable for use at temperatures 
of (a) — 20()°C. : 
(London, B.Sc.) 



(b) — 50*G„ (c) 50°C., (d) 700°C> (e) 200 a C. 



♦These examples have been classified and arranged chapterwJse correspond- 
ing to the twelve chapters of the book. 



332 



EXAMPLES 



11. Describe two methods for measuring high temperatures. 
State: clearly the principles underlying them and the range and sensiti- 
vity of each. (A, V» B. St, 1941.) 

12. Write a short essay on the measurement of (a) high and 
(6) low temperatures. (Delhi Univ., 1954; Punjab Univ., 1946.) 

iS. In what respects is a constant volume gas thermometer 
superior to the constant pressure gas thermometer and the mercury- 
in-glass thermometer. Describe the construction and use ot the inter- 
national standard hydrogen constant volume thermometer. (Patna 
Univ., 1949; Utkal Univ., 1954,) 

II 

1. Enunciate Newton's law of cooling and show how corrections 
can be made for the heat lost by radiation during calorimetric experi- 
ments. Establish a relation for finding the specific heat, of liquids by 
the method of cooling. (Nagpur, B.'Sc.) 

2. If a body takes B minutes to cool from 100°C. to 60 C C„ how 
long will it take to cool from 60°C> to 20*C, assuming that the 
temperature of the surroundings is 10°C, and that Newton's law 
•of cooling is obeyed. 

3. Describe Joly's differential steam calorimeter and explain 
how it. is used for finding the specific heat of a gas at constant volume. 
State the corrections to be made. (Punjab Univ., 1952, 1954 ; 

jerat Univ., T 95 1 ; Patna Univ., 1947; Calcutta Univ., 1947.) 

4. Describe the steam calorimeter. Explain how it may be 

letermine (i) the specific heat of a gas at constant volume, 
rific heat of a small solid. (A. U., 13. Sc.) 

5. Describe a method of determining the specific beat of a gas 
at constant volume, giving a neat diagram of the arrangement of the 
apparatus necessary. Why is the specific heat at constant pressure 

ter than that at constant volume ? (Dacca, B.Sc.) 

6. Describe the constant flow method of Callendar and Barnes 
for the measurement of the mechanical equivalent of heat. In an 
experiment using this method, when the rate of flow of water was 

i, per minute, the heating current 2 amperes and the difference 
of potential between the ends of the heating wire 1 volt, the rise 
of temperature of the water was 2-5°C, On increasing the rate of 
flow to 25.4 gm, per minute, the heating current 3 amperes and 
the potential difference between the ends of the heating wire to 1-51 
volts, the rise of temperature of the water was still 2.5 °G. Deduce 
the value of the mechanical equivalent of heat. (A. U., B.Sc.) 

7. Describe Nernst vacuum calorimeter and indicate briefly how 
it has been used for measuring specific heats at low temperatures. 
(A. U., M.Sc, 19,25,) 

8. Give an account of the continuous flow method of measuring 
the specific heat of a gas at constant pressure and point out its 
advantages. (A. U., B,Sc, 1938.) 

9. Describe an accurate method of measuring the specific heat 
of a gas at constant pressure. (Punjab Univ., 1941; Gujerat Univ., 
1949, 1951.) 






EXAMPLES 



333 






'-'; 



10. In a determination of the specific heat at constant pressure 
r Regnault's method the gas is supplied from a reservoir whose 

volume is 30 litres at 10°C. The pressure of the gas in the beginning 
is 6 atmos. and in the end 2 atmos., the temperature remaining 
constant at 10°C. The gas was heated to I50°'C. and led into a 
calorimeter at 10-0°C. The final temperature of the calorimeter and 
contents was 31-5°C. and its water equivalent was 210 gm. If the 

ly of the gas is 0*089 gm. per litre at N. T. P., calculate its. 

c heat at constant pressure. 

11. A quantity of air at normal temperature is compressed 
slowly, (b) suddenly, to j\, of its volume. Find the rise of 



(a 

temperature if any; in each case 

[Ratio of the two specific heats of air = 1*4, log 2-75 == -4362 ; 

log 6-858 — 0'83fi2.] 

Deduce the formula used for (b) . (A. U-, B.Sc.) 

12. In a Wilson apparatus for photographing the tracks of 

particles the temperature - 



of the air is 20°C. li its volume is in- 



creased in the ratio 1-375:1 bv the expansion, assumed adiabatic, 

calculate the final temperature of the air. (The ratio of specific heats 
of air = 1-4L) 

15. Distinguish between adiabatic and isothermal changes and 
show that for an adiabatic change in a perfect gas jbu *= constant 
where y is the ratio of specific heat at constant pressure and constant 
volume respectively. (Allahabad Univ., 1952; Punjab Univ., 1954, 

9 no.) 

14. Deduce from first principles the adiabatic equation of a 
perfect gas. 

A motor car tyre is pumped up to a pressure of two atmospheres 
at 15°C, when it 'suddenly bursts. Calculate the resulting drop ir 
the temperature of air. (A. U., B.Sc, 1938.) 

15. Describe a method of determining the ratio of the specific 
heats of a gas at constant pressure and constant volume. How has 
the mechanical equivalent of heat been calculated from a k 

of this ratio? (A. U-, B.Sc.) 

16. Explain why the specific heat of a gas at constant pressure 
is greater than that at constant volume. Obtain an expression for the 
difference between the values for a perfect gas. Find the numerical 
value of their ratio for monatomic gases. (Punjab Univ., 1949, 1951; 
Utkal Univ., 1952.) 

17. Describe the Clement and Dcsonncs' method of finding the 
ratio of specific heats of air, giving the simple theory of the method. 
What are the objections to the method and whnt modifications and 
improvements have been proposed. (A, U., 1950; Bihar Univ., 1954; 
Utkal Univ., 1955; Punjab Univ., 1953, 1958.) 

18. Derive the relation between volume and temperature of a 
mass of perfect gas undergoing adiabatic compression. 

A quantity of dry air at 15°C is adiabatically compressed to -J-th 
of its volume/ Calculate the final temperature given y — 1-4 and' 
4*'* = 1-74 (Aligarh Univ., 1948; Punjab Univ., 1957.) 



334 



F.XAMl'U'5 



19. Explain how the mechanical equivalent of heat can be 
deduced from a knowledge of die specific heats of air at constant 
pressure and constant volume. State clearly any assumptions made 
in your reasoning and describe experiments, if' any, -which afford 
justification for such assumption. (Madras, B.Sc.) 

20. Describe a method of determining the ratio of the two speci- 
fic heats of a gas. Show that it follows from the kinetic theory of 
gases that the ratio of the two specific heats in the case of a mona- 
tomic gas is 1.66, (Madras, B.Sc.) 

21. Find the ratio of the specific heats of a gas from the follow- 
ing data : — A flask of 10 litres capacity weighs, when exhausted, 
160 gm. ; filled with the gas at a pressure of 75 cm, of mercury it 
weighs 168 gm. The column of the gas, which when confined in a 
tube closed at one end and maintained at the same temperature as 
the gas in the flask, responds best to a fork of 223.6 vibrations per 
second, is 50 cm. 

22. Determine the ratio of the specific heats of air from the 
following data :— Velocity of sound =34215 cm. per sec. in air at 
750 mm, and 17°C ; density of air — ,00129 gm. per c.c, at N. T. P, ; 
coefficient of expansion of air— ^ £=981 cm. /sec. 2 j density of 
mercury e= 13.6 gm, per c.c, (Manchester, B.Sc.) 



Ill 



I. 



In an experiment with Joule's original apparatus the mass of 

the ; on either side was 20 kilograms and each fell through a 

one metre forty times in succession. The water equivalent 

calorimeter and its contents was 6 kilograms and the rise in 

temperature during the experiment was 0.62°C. Calculate the value 

of the mechanical equivalent of heat, 

2. Determine Lhe heat produced in stopping by friction a fly- 
"' kilograms in mass and 50 cm. in radius, rotating at the rate 

ond assuming the fly-wheel to be a disc mounted 
axial b and having a uniform distribution of mass, 

3. A canon ball of 100 kilograms mass is projected with a velocity 
of 400 metres per second, Calculate the amount of heat which would 
be produced if the ball were suddenly stopped. 

4. In one hour a petrol engine consumes 5 kilograms of petrol 
whose calorific value is 10,000 cals. per gram. Assuming that h per 
cent of the total heat escapes with the exhaust gases and that 12 per 
cent of the heat is converted into mechanical energy, find the average 
horse-power developed by the engine and die initial rate of rise of 
temperature of the engine per minute. Radiation losses mav be 
ignored and the water equivalent of the whole engine is 40 kilograms. 

5. Give the outlines of the methods bv which the mechanical 
equivalent of heat can be determined. Assuming- that for 



mi no 



air at 



constant pressure the coefficient of expansion is 1/273, the density 
at C. and atmospheric pressure is 0,001293, the specific heat. 






-fc 



EXAMPLES 



335 



^==0.2389 and the ratio c p fe s = 1.405, calculate the mechanical 
equivalent o£ heat. Suppose that there is inappreciable cohesion 
between the molecules. (Bombay, B.Sc.) 

6. Define the mechanical equivalent of heat. If the kinetic 
energy contained in an iron bail, having fallen from rest through 
21 metres, is sufficient to raise its temperature dirough 0.5 °C V 
calculate a value for the mechanical equivalent of heat (given g = 980 
cm. per sec. per sec. and specific heat of iron = 0J) . 

7. Show how the method of electrical heating has been adopted 
in the determination of the mechanical equivalent of heat. One 
gram of water at 100°C. is converted into saturated vapour at the 
same temperature. Calculate the heat equivalent of the external work 
done during the change. Density of water at 100°C. = 0.958 gms. 
per c.c. ; density of saturated steam at 100°C. = 0.000598 gms. per c.c. 

8. Describe a laboratory method of determining J. Give expe- 
rimental details. (Delhi Univ., 1951.) 

9. Describe with relevant theory Rowland's method of finding 
the mechanical equivalent of heat. Point out the significance of the 
result. (Nagpur Univ., 1956.) 

10. The height of the Niagara Falls is 50 metres. Calculate 
the difference in temperature of the water at the top and bottom 
of the Fall if J — 4.2 X 10 T ergs/cal. 

11. A lead bullet at temperature of 47.6°C strikes against an 
obstacle. If the heat produced by the sudden stoppage is sufficient 
to melt the bullet, with what velocity the bullet srikes the obstacle ? 
It is assumed that all the heat goes to the bull el. Melting point of 
lead = 327°C, specific heat of lead = .03 i ./degree, latent heat 
of fusion of lead = 6 cal./gm. f J = 4.2 X 10 T ergs/cal. (U ileal Univ., 
1 950,) 

12.^ Describe Callendar and Barnes' continuous flow method of 
measuring the mechanical equivalent of heat. State how the method 
can be adopted to measure the variation of specific heat of water 
between 10 Q C and 90 C 'C. (Punjab Univ., 1951 ; A. U., 1951.) 

i:>. Explain what is meant by the "velocity of mean square" of 
molecules of a gas and their "mean free path". Show how thsse 
two quantities can be found. (Bombay, B.Sc.) 

14. Calculate the molecular velocity (square root of the mean 
square velocity) in die case of a gas whose density is 1.4 gm. per litre 
at a pressure of 76 cm, of mercury. Density of mercury — 18.6, 
g=:981 cm. per sec, per sec. (Manchester, B.Sc.) 

15. Show that pressure of a gas is equal to two-thirds of the 
kinetic energy of translation per unit volume. Calculate the kinetic 
energy of hydrogen per gram-molecule at 0°C. (A. U., B.Sc, 1949) . 

of gases 

this theory ? (A. U., B.Sc. 



Deduce Boyle's and Avogadro's laws from the kinetic theorv 
. What interpretation of temperature is given according to 



17. Outline the essential features of the kinetic theory of gases 
*md an expression for the pressure of a gas on the basis of kinetic 
theory. (Punjab Univ., 1954, 1955, 1957; Delhi Univ., 1954.) 



336 



EXAMPLES 



18. Show that the pressure exerted by a perfect gas is | of the 
kinetic energy of the molecules in a unit volume. Explain on the 

basis of the 'kinetic theory (i) why the temperature of a gas rises 
when it is compressed, and (ii) why the temperature of an evaporat- 
ing liquid is lower than its surroundings, (Punjab Univ. r 1956.) 

19. Deduce an expression for the conductivity of a gas from the 
kinetic theory. How would you actually proceed to determine the 
conductivity of any particular' gas ? (A. U„ B.Sc, lions,, 1931.) 

20. State the law of equipartition of energy. Prove that for a 
monatomic gas, the value of gamma, the ratio between the specific 
heats is 5/3 and for a diatomic gas it is 7/5. (A, IL, B.Sc., 1932,) 

21. l r ind an approximate expression for the mean free path of 
a molecule in a gas, and give a short account of any one phenomenon 
depending on the length of the mean free path. (London, B.Sc., 

Horn.) 

22. On the basis of the kinetic theory deduce an expression 
for the viscosity of a gas in terms of the mean free path of its mo 
cules. Show that it is independent of pressure but depends upon 
the temperature of the gas. (Baroda Univ., 1954,) ^ 

23. What is meant by (a) the "coefficient of viscosity" of a gas, 
he ri-'. free path of its molecules"? Show how to deduce 

a relation between these quantities from the kinetic theory, (Lond., 
B.Sc.') 

24. Describe some phenomena which have led to the conclusion, 
that molecules have a finite diameter and mean free path. How can 

atter be determined? 

IV 

1. How has van der Waals modified the isothermal equation for 
a gas ? Calculate the values of the critical pressure, volume and tem- 

rature in terms of the constants of his equation.. How do the 
theoretically derived results tally with experiments? (A. U„ B-Sc, 
193.1.,) 

2. Derive van der Waals" equation of state and obtain expres- 
sions for the critical temperature, pressure and volume in terms of 
the constants of van der Waals' equation. (Punjab Univ., 1957 ; 
Delhi Univ., 1953 ; Bombay Univ., 1953 ; Patna Univ., 1948 ; Calcutta 
Univ., 1948 : Aligarh Univ.. 1950 ; Nagpur Univ., 1953 ; Baroda 
Univ., 1954.) 

3. Express the value of the critical temperature in terms of a, 
b and R, Calculate its value for CO... where a — .00874 and h = .0023. 
(Punjab Univ., 1957 ; Allahabad Univ., 1952.) 

4. Define critical temperature, pressure and volume of a vapour 
and give some account of the behaviour of a substance near the criti- 
cal point. (A. U., B.Sc.) 

5. Draw a diagram showing the general form of the isothermal 
including both liquid and vapour state, and explain the meaning of 
the different parts of the curve. What is the true form of the straight 
portion of this curve and why ? (A, U., B.Sc, 1935.) 



rXAMPLES 



m 



6. Give an account of the properties of fluids in the neighbour- 
hood of the critical point. Describe bow you would determine the 
critical constants of a substance, (A. IL, B.Sc., Hons,, 1928). 

7. What is meant by the critical point in the state of a fluid ? 
Show on a diagram the character of typical isothermals of a fluid 
above and below the critical temperature. Explain how van der 
Waals' equation accounts for the existence of a critical point. 
(London, B.Sc.) 

8. Explain how van der Waals' equation accounts for the exist- 
ence of a critical point. Calculate the values of tile critical pressure, 
critical volume and critical temperature for a gas obeying van der 
Waals' equation. (A, U., B.Sc, 1947.) 

9. Describe the experiments of Andrews on carbon dioxide. 
State and discuss the results obtained by him. Hence show that 
liquid and gaseous states are only distant stages oE a long series of 
continuous changes. (Bombay Univ., 1953 : Baroda Univ., 1954 ; 
Punjab Univ., 19 

10. Explain in brief outline the reasons which led van der 
Waals to his equation {p + afv*) (v — b) =RT. Discuss how far 

\i equation is in keeping with' experimental facts. (Allahabad 
Univ., 1950.) 

IL Describe Arnagat's experiments on die compressibility of 
gases at high and low pressures. State his important conclusion*. 
Indicate how the results can be explained with the help of the van 
der Waals' equation. (Madras Univ.) 

12. Derive the reduced equation of a gas starting from van der 
Waals' equation of state. Show that if two gases have the same 
reduced pressure and volume, they also have the same reduced tem- 
perature, (Punjab Univ.) 

V 

1. How would you determine the vapour pressure of a liquid 
above its normal boiling point ? Explain how clouds are formed by 
the mixing of warm moist air with cold moist air, (A. U., B.Sc) 

2. Describe a method of determining the vapour density of a 
volatile liquid, and explain the theory of your method. (Dacca, B.Sc.) 

Give an account of a method which has been adopted for 
the determination of the pressure of saturated vapour between I00°C,. 
and I2(PC, Explain clearly what is meant by the statement that 
the specific heat of saturated vapour at 100°C is negative. /Madras 
B.Sc.) v 

4. An electric current of 0.75 ampere is passed for 30 minutes 
through a coil of wire of 12.4 ohms resistance immersed in benzene 
maintained at its boiling point, and 29.85 gm. of benzene are found 
to have vaporised. Calculate the latent heat of vaporisation of 
benzene. 

5. What are the principal differences between saturated and 
unsaturated vapours? How would you determine the pressure of 
saturated water vapour at temperatures between 40 °C and 11.0°C? 



338 



EXAMPLES 






859 






6, A mixture of a gas and a saturnLcd vapour is contained in 
a closed space. How will the pressure of tJie_ mixture vary 
(a) when the Leinperature is changed and Lhe volume is kept constant, 

hen the volume is changed and the temperature is kept constant ? 

7, Describe and discuss a method by which you could determine 
the latent heat ol a metal which melts at about 20Q e C 

8, Define latent heat of evaporation ol liquids and describe . 
how it can be measured accurately. Stale Trouton's law connecting 

atenl Nat of evaporation with the absolute temperature of the 
boiling point. 

VI 

1. Write an essay on 'artificial production of- cold'. (A. U., 
B.Sc, Hons., J 030.) 

2. Describe the manufacture of liquid air. (A. U., Ji.Sr,, 1030.) 

3. Discuss theoretically the production of cold by expansioi 
gases through porous plugs. I low has iiuriple bee; 

in m:: or liquefying air? (A. U., B.Sc, 1937; Dell 

1953, 1954, 1959.) 

4. Write a short essay on the liquefaction of the so-called per- 
manent gases. (Dacca, B.Sc.) 

5. ig hydrogen, and explain the 
principle involved in the process. (A. u., B.Sc.) 

6. Describe and discuss the porous plug experiments of Joule 
and Kelvin. Explain what is meant 

ice to hydrogen and h< 

ii the meth; in the manufacture 

of I '.Sr., 193 

8. Dii tiabatic change and Toule- 

Thomsoii cribe hov. ier has been utilised for I 

\. V., B. 7.) 

!i. •.: porous plug experiment of Joule and Thomson 

and discuss the result ed tor differed h special refer- 

ence to hydrogen. Indicate hov.' the results have been utilised for 
the Hi ►! ail and h] (Bihar Univ., 1951; Punjab 

Univ., 19! 

10. What Is Joule-Thomson effect? Obtain the expression for 
iced assuming that the gas obeys van de 
equation. Wlr, do hydrogen and helium show a heating 
ordinary temperature? (Allahabad Univ., 1955; Rajasthan Univ., 
I960.) ' 

VII 

1. Describe and explain a method of measuring the linear 
expansion of solids by means of interference bands. 

2. If a crystal has a coefficient of expansion 13 X 1G"" T in one 
direction and of 231 X 10- T in every direction at right angles to the 
first, calculate its coefficient of cubical expansion. (London, B.Sc.) 



3. A lump of quartz which has been fused is suspended from a 
quartz fibre and allowed to oscillate under the influence of the torsion 
of the fibre. If the coefficient of linear expansion of the material Is 
7 X 10- T and the temperature coefficient of its rigidity is -f- 13 X 1®~S 
how many seconds a day or what fraction of a second a day would a 
nge of temperature of 1°C1 make? (London, B.Sc.) 

•1. A seconds pendulum is one which completes half an oscilla- 
tion in I second. Such a pendulum of invar is given and is correct 
at I0*C. If the average temperature for the three months of June, 
July and August is 25°C. and the clock is correct at 12.0 a.m. on 
June ]st, how much will it be incorrect at 12.0 a.m. on September 
1st ? Coefficient of expansion of invar is 1 X 1Q - *** 

5. Describe, in full detail, the method by which the expansion 
of crystals, when heated, may be studied experimentally. (Allahabad 
Univ., 1943; Aligarh Univ.. 1949; Punjab Univ., 1946, 1954.) 

6. Describe a method by which the cubical expansion of a liquid 
can be accurately determined by weighing a solid of known expansion 
in it at two known tempi 

A solid is found to weigh 29.9 gms. in a liquid of specific gravity 
' T:. its weight in air bei gms. It we3 

in the same liquid at 25°C pecific gravity is 1.17. Calculate 

the coefficient of cubical ol the 

7. («) Describe K method for the determination oi 
coefficient of absolute expansion of ,. Indicate briefly the pre- 

■ led by Regnault to avoid errors. 

(h) If the coefficient ol cubical expansion of glass and mercury 

1 and 1,8 what fraction of the whole 

volume of a Id be filled with mercury in order that 

the empty p a it should re i nsta n t when gl ass and 

ed to the same i mre. (Dacca, B.Sc.) 

S. Describe and explain the interference method For finding 
the coefficient : expansion of crystals. Ii 1 ss the 

rial expansion of a crystal differ from that of an c metal, 

(Punjab Univ., 1961.) 

9. Describe I' s method of finding the absolute co- 

efficient of expansion of nn te calculations 3 

in it. Why is this method better than others? (Calcutta Univ., 
1949.) 

1 ii The difference between the fixed points of a mercury therm o- 

is 18 cm. If the volume of the bulb and capillary tube up to 

the 0° mark is 0.0 c.c, calculate the sectional area of the capillary. 

■•Jem of cubical expansion of mercury == T.S X MH*> coefficient 
of lii mansion of glass — 7 X JO -6 . (Madras Ur 

11. An air bubble rises from the bottom of a pond, where 
the temperature is 7°C, to the surface 7 metres above, at which the 
temperature is 17°C. Find the relative diameters of the bubble in 
the two positions., assuming that the pressure at the pond surface is 
equal to that of a column of mercury of density 13.6 gm. per cc. 
and 76 cm. in height. 



340 



EXAMPLES 



VIII 

1. By what processes does hot water in an open vessel lose 
heat ? Describe experiments by which the several causes of loss may 
be shown to exist. (Dacca, B.Sc.) 

2. Define the thermal conductivity of a substance and describe 
some way of finding it. 

An iron boiler 5/8 inch in thickness exposes 60 square feet of 
surface to furnace and 600 lbs. of steam at atmospheric pressure are 
produced per hour. The thermal conductivity of iron in inch-lb.-scc. 
Slits is 0.0012 and the latent heat of steam is 536. Find the tem- 
perature of the underside of the heating surface. Explain why this 
is not the temperature of the furnace. (Dacca, B.Sc.) 

3. Explain the difference between the thermal conductivity and 
the diffusivity of a substance. 

The two sides of a metal plate 1.5 square metres in area, and 
0.4 cm. in thickness are maintained at 100°C. and 30 D C. respectively. 
If the thermal conductivity of the metal be 0.12 C.G.S. units find 
the total amount of heat that will pass from one side to the other 
in one hour. (Dacca, B.Sc.) 

4. The interior of an iron steam-pipe, 2.5 cm. internal radius, 
carries steam at 140 C C. and the thickness of the wall of the pipe is 
3 mm. The coefficient, of emission of the exterior surface (heat lost 

econd per sq. cm. per degree excess) is 0.0003 and the tempera- 
ture of the external air is 20 C C. If the thermal conductivity of iron 
is 0.17 C. G. S. unit find the temperature of the exterior surface, and 
bow much steam is condensed per hour per metre length of tube, 
a tent heat of steam at 140°C, being 509. (London, B.Sc.) 

5. The thickness of the ice on a lake is 5 cm. and the tempe- 
of the air is - 10°C. At what rate is the thickness of the ice 

dmately how long will it take for the thickness 
■ oubkd ? 
(T ■; conductivity of ice == 0.004 cal. cm- 1 sec-* °C~\ Density 

of ice — 0.92 gm. per c.c' Latent heat of ice = 80 cal. per gm.). 

6. Distinguish between thermal conductivity and therm ometric 
conductivity of a substance. Describe a method of finding the 
thermal conductivity of a solid. 

Calculate the rale of increment of die thickness of ice layer on 
a lake when the thickness of ice is 20 cm. and the air temperature 
is -40°C. Thermal conductivity of ice = 0.004 cal. cm.- 1 
"^C- 1 , density o£ ice = 0.92 gm./c.c. and its latent heat = 80 caL/jgffl. 
After what time the thickness will be doubled ? (Bihar Univ., 1954.) 

7. Define the terms conductivity and diffusivity as used in the 
thcory of heat conduction. Describe a method of comparing the con- 
ductivity of two metal bars. Account for the fact that the evapora- 
tion of liquid air is greatly reduced when kept in a Dewar vacuum 
vessel. (Madras, B.Sc.) 

8. Define conductivity. Deduce expression for the flow of heat 
in a long; bar when it has acquired a steady state. (B. H. U., B.Sc, 
1931J 



EXAMVLliS 



Ml 



9. Define the coefficient of heat conductivity of a substance 
and give details of some method of determining this constant for 
iron.'"' (A. V., B.Sc. 1931.) 

10. Describe lngen-Hausz's experiment, and prove from the 
mathematical theory that the conductivities of different bars vary as 
the square of the length up to which wax is melted. (A. U., B.Sc, 
1945, 1950.) 

IL Distinguish between thermal conductivity and thermometrie 
conductivity- Bring out, the connection between the two. Describe 
a tie's method of determining the thermal conductivity of a solid. 
(Punjab Univ., 1950.) 

VI. Show that in the steady state of a metal bar heated at one 
end 

Ifr = *** 
■where the symbols have their usual significance. 

Hence prove that the length to which the wax melts in the steady 
state along a wax coated bar is proportional to the square root of the 
coefficient'of thermal conductivity of the material of the bar. (Allaha- 
bad Univ., 1955.) 

13. Describe Forbes' method of determining- the thermal con- 
ductivity of a metallic bar, and explain the formulas used. (Lucknow 
Univ., 1950 ; Allahabad Univ., 1951 ; Punjab Univ., 1954, 1955, 1957 J 
Utkal Univ., 1953.) 

14. Define thermal conductivity, A steady stream of water 
flowing at the rate of 50 grams a minute through a glass tube 30 cms. 
long, 1 cm. in external diameter and 8 mm. in bore, the outside of 
which is surrounded by steam at a pressure of 700 mm., is raised in 
temperature from 20°C. to S0 a C. as it passes through the tube. Find 
the conductivity of glass. You are given that log 1.25 = 0.223. 
Deduce anv formula that you use. (A, IL, B.Sc, 1937 ; Patna Univ., 
1948.) 

15. Define thermal conductivity. Describe a method you have 
adopted in experimentally finding this constant for a good conductor. 
Find the coefficient of conductivity of a badly conducting material 
upon which, the following experiment was made ;— A very thin- walled 
hollow silver cylinder 40 cm. in diameter and 50 cm. in length is 
covered all over its external surface including 1 the ends by a layer of 
the material 0.33 cm. in thickness. Steam at a temperature of 100°C 
is passed through the cylinder and the external temperature is 20°G 

r is found to accumulate within the cylinder at the rate of 3 gm. 
per minute. The latent heat of vaporization of water at 100°C. is 
537 cal. per gram and cross-sections of the steam inlet and outlet are 
each 10 square centimetres. (Madras, B.Sc.) 

16. Define conductivity and diffusivity of hcaL When steam 
is passed through a circular tube of length I and having the internal 



342 



JCXAMPIJES 



and external diameters a and b respectively, prove that the radial flow 
of heat outwards is given by 

2irKl((t x -e 2 )/lag e ~ 

where K is conductivity and $ t , 0. 2 the temperatures inside and out- 
side the tube, How will you determine the conductivity of india- 
rubber? (A, U., B.Sc„ lions., 1930.) 

17. Steam at 100°C. is passed through a rubber tube, 14.1? cm. 
length of which is immersed in a copper calorimeter of thermal capa- 
city 23 cat, containing 440 gm. of water. The temperature of the water 
and calorimeter is found to rise at the rate of 0.019 d C. every second 
when they are at the room temperature (22 °C.) . The external and 
the internal diameters of the tube are 1,00 cm, and 0,75 cm. respec- 
tively. Calculate the conductivity of india-rubber. 

18. Prove Lhat if a long bar is periodically heated at one end, 
the law of propagation of heat is given by the equation 

dd _ _^0 

" * iP ' 

Obtain a solution of this equation and show how with its aid the 
diurnal and annual variation oh temperature at some depth below the 
surface of the earth can be explained. (A, U., M.So, 1926.) 

19. Heat is supplied to a slab of compressed cork 5 cm. thick 
and of effective area 2 sq. metres, by a heating coil spread over its 

ace. When the current in this coil is 1.18 amp. and the potential 

difl s its ends 20 vol is, the steady temperatures of the 

E the slab are 12.5°C. and 0°C. Assuming that the whole of 

the heat developed in the coil is conducted through the slab, calculate 

conductivity of the cork. 

Define coefficient of thermal conductivity and describe Lees' 

hod for determining the thermal conductivity of metals. Equal 

jer and aluminium are welded end to end and lagged. 

If i 1 ends of copper and aluminium are maintained at 100 o C 

and 0°C respectively, find the temperature of the welded interface. 

Assume the thermal conductivity of copper and aluminium to be 

0.92 and 0.50 respectively. (Punjab Univ., 1953, 1956, I960.) 

21. Describe Lees' method of determining the thermal conduc- 
tivity of a bad conductor, (Bombay Univ., 1947.) 

--. Describe and explain the cylindrical shell method of di 
mining She conductivity of a solid. (Bombay Univ., 1948.) 

23. Describe and give the theory of a method useful for a 
practical determination of the thermal conductivity of a liquid. 
(London.. B.Sc.) 

24. Discuss some methods by which the thermal conductivity 
of a gas has been determined. What are the experimental difficult.': 
and how have they been overcome ? 

25. How is the thermal conductivity of a liquid determined ? 



EXAMPLES 3*3 

How do you demonstrate that hydrogen is more conducting than 
air? (A. U., B.Sc, IS 

20, Discuss the difficulties which beset the investigation of the 
thermal conductivity of gases and indicate how and to what extent 
they have been overcome, (London, B.Sc.) 

IX 

I. What do you understand by a reversible cycle as opposed to 
an irreversible one? Give instances of each, (A. U., B.Sc., 1931.) 

What is meant by a reversible change? Describe Carnot's 
cycle and prove that the efficiency of all reversible engines working 
between the same two temperatures depends only on the temperature 
of the hot and cold bodies. (Bombay Univ., 1947, 1948, 1949 ; Lkkal 
Univ., 1954; Punjab Univ., 1953.) 

3. Describe Carnot's cycle and prove Carnot's theorem. What 
•nt by a reversible change ? State briefly how Carnot's theorem 

h to an absolute scale. (Delhi Univ.., 1952, 1954). 

4. State and explain the significance of the second law of thermo- 
dynamics. Show that the efficiency of a reversible engine is maximum. 

iiv., 1950 ; Punjab Univ., 1957.) 
Describe a Diesel engine and deduce an expression for its 
efficiency. Can the Carnot engine be realised in practice? (Punjab 
Univ., 1949.) 

fi. Describe with diagrams an Otto engine and deduce an ex- 
ion for its efficiency. (A. U., B.Sc, 1949.) 

7. Describe some kind of internal combustion engine and explain 
fully how heat is thereby converted into work. 

Describe its uses and applications and dwell upon its advantages 
over the steam engine. (A. U., B.Sc.) 

8. Describe the cyclical process of a steam engine and compare 
its efficiency with that of an internal combustion engine. Explain 
(1) why steam engine is preferred in railways, and (2) why petrol 
engine is used in. aeroplanes. (A. U,, B.Sc, 1930.) 

9. Write an essay on 'Heat engines'. (Dacca, B.Sc) 

10. Explain the indicator diagram and apply it to Carnot's cycle. 
Explain the conditions for reversible working and show that in ge 

for reversible cycles 1 -y=0. (Bombay, B.Sc) 

I I . Describe Carnot's cycle. Show how the work done during 
each operation is represented on a pv diagram. Find an expression 
for the Work done during ouch operation when the working substance 
is a perfect gas, (London, B.Sc.) 

12. Discuss the statement : 'Reversibility is the criterion of 
perfection in a heat engine'. Explain with two examples what vou 
understand by a reversible process. 

A Carnot engine works between the two temperatures I00°C. and 
10*C. Calculate its efficiency. (A. U., B.Sc, 1947.) 



344 




EXAM 



X 



EXAMPLES 



345 



1. Show that when a body expands, the external work performed 
is given by the expression 






From the following data calculate what fraction of the specific 
heat of copper is due to the external work done in expansion in an 
atmosphere at a pressure of 76 cms. of mercurv :— 

Specific heat of copper = 0.093, Specific gravity of copper = 8,8, 
of mercury = 13. 6. Coefficient of linear expansion of copper = 0.00 On 1 6, 
J = 4,2 X 10 7 ergs per calorie. (A. U., B.Sc) 

2. Deduce an expression for the work required to compress 
adiabatically a mass of gas initially at volume v x and pressure pi to 
volume v 2 . 

Find the work required to compress adiabatically I gm. of air 
initially at N.T.P. to half its volume, Densitv of air at N.T.P, 
=== 0.00129 gm./c.c. and c p jc =z 1.4. (Birm,, B.Sc.) 

3. Explain external and interna! work. Discuss the changes 
in the kinetic energy of the molecules of a gas when heated and 
hence show that the ratio of the specific heats for a monatomic eai 
is 5/3. * 

4. Write an essay on the transformation into heat of other forms 
of energy. (Cal, B.Sc.) 

5. Tn what sense can the second law of thermodynamics be 
ded as furnishing ail absolute scale of temperature ? How can 

readings of gas therm i be reduced to this particular scale ? 

!J.; B.Sc, Bom., 1931.) 

-i what you .i id by a thermodynamic scale of 

rature. Show that it agrees math an ideal uas scale. (Punjab 
Univ., 1053, 1954, 1956, 1957; Bombay Univ., 1953.) 

7. Write notes on the following : — 

(a) Lord Kelvin's absolute scale of temperature, 

(b) Joule and Kelvin's porous plug experiment, 

' B.Sc, 1930.) 

8. How did Kelvin arrive at the absolute scale of temp 
Show that the ideal gas scale and the absolute scale are Iden- 
[Iow is the absolute scale I in practice ? (Delhi Univ., 



(Nagpur, 



ture ? 
deal. 



1951, 1953, 1956.) 

9. Define a scale oF temperature without malting use of the 

peculiarities of any selected thermometrit substance. Show (a) that 
Kelvin's work scale is such a scale, and (b) that the ratio of two tem- 
peratures as measured on the Kelvin scale is identical with the ratio 
of the same two temperatures on the perfect gas scale. (London, 

10. Explain what you mean by die entropy of a substance. 
Show that for any reversible cyclic change of a system the total change 






of entropy is zero. Explain why this statement is not true for irrever- 
j changes. (Punjab Univ./ 1958; Baroda Univ., 1955.) 

11. Explain the idea of entropy. Derive an expression for the 
entropy of m grams of perfect gas. (Allahabad Univ., 1950.) 

12. A volume of a gas expands isotherm ally to four times its 
initial volume. Calculate" the change in its entropy in terms of the 
gas constant. (Baroda Univ., 195C) 

13. Calculate the change of entropy when 100 gms. of water at 
S0 Q C. are mixed with 200 gms. of water at 0° assuming that the 
specific heat of water is constant between these temperatures. 
(London, B.Sc.) 

14. Derive an expression for the entropy of a perfect gas in 
terms of its pressure, volume and specific heats. (Bombay, B.Sc.) 

15* Prove that the increase of entropy per unit increase of 

volume under constant pressure is equal to the increase of pressure 
per unit increase of temperature during an adiabatic change. (Bombay, 
B.Sc.) 

16. Calculate the change of entropy when 10 gm, of steam 
at 100°C, cools to water at 0°, assuming that the latent heat of 
vaporisation is 536 and the specific heat of" water is 1 at all tempera- 
tun. 

17. State the second law of thermodynamics and apply it to 
the determination of the effect of pressure on the melting point of a 
solid. (A, U., B.Sc) 

15. Give an elementary proof of Clapeyron's relation 

dp L 

dT 1 (r s — v L Y 
Discuss how the boiling- point of a liquid and the melting point 
■a solid are affected by change of pressure. (Punjab Univ., 1917, 
1957.) 

19. Derive Clapeyron's equation 

dp L _ 

dT 7{p 2 — v t ) ' 

Calculate the change in temperature of the boiling point of 
water due to a change of pressure of 1 cm. of mercury. (L = 536 
calories, volume of 1 gm. of water at 100°C = 1 c.c, volume of 1 gm. 

or saturated steam at 100°C — 1600 c.c.) . (Delhi Univ., 1955.) 

20. Prove the thermodynamic relation 

>a\ tdp\ 

\ dv Id ' \dOL 



and hence prove 



ft'0 



(*-*) 



(B. H. U„ B.Sc, 1931.) 



346 



KXAMPi.KS 



21. Deduce the latent heat equation 

Calculate the depression of the melting point of ice per atmos- 
pheric increase oL" pressure, given latent heat of fusion — bO cal. and 

njity of ice at 0°C. 10 ~, „ c ir iar.\ 

r nor -, — rr- (Nagpur, B,Sc„ 1930.) 

density of water at O'C. 11 v bl l 

22. Prove that 



One gram of water-vapour at 100°C, and atmospheric pressure occu- 
pies a volume of 1640 c.c. and L — 53G calories, Prove that the 
vapour pressure of water at 99°C. is about 733 mm, of mercury. 
(A, U., RlSc, 1926.) 

23. If sulphur has a specific gravity 2.05 just before, and 1,95 
just after melting, the melting point being 115°G. and the latent 
heat 9.3, find the alteration in melting point per atmospheric eha 

of pressure. (I.ond., B.Sc, Hons.) 

24. Calculate the von I me of n gramme of steam at 100°C, given 
that the lati i of evaporation of water at 10G°C = 535 and 
the change of boiling point is 0,37°G. per cm. of mercury pressure 
(/ £=4.2X10*). (Manch., B.5c.) 

25. Calculate £ given that the change of 
of one phere changes the melting point of ice by 

C and when one gram of ice melts volume changes by Q.O907 
(Punjab t 

26. Discuss the effect of change of pressure on the polling point 
of a liquid. 

olum.es of water and saturated steam 
at 1Q0°C are 1 c.c. and 1601 c.c. respectively and the latent heat 
vaporisation is 536 cal./gm., find the change in boiling point for a 
change of pressure of 1 cm. of mercury. (Allahabad Univ., 1946 j 
Calcutta Univ., 1948.) 

27. Show, on thermodynamic principles, that when a liquid 
film is suddenly expanded, it must fall in temperature, and find an 
expression for the lieat that must be supplied to keep it constant. 
(A. U., B.Sc, Hons., 1927.) 

28. Deduce an expression for the specific heat oi saturated 
vapour and prove that in the case of water at 100°C., It is negative. 
How do you explain the paradox 'negative specific heat ?' (A. U., 
B.Sc., Hons., 1931.) 

29. Define the 'triple point*. Describe the successive changes 
observed in a system containing water at its various states when 
pressure is changed at a constant temperature (a) when the tern- 

■e is above the triple point, (b) when the temperature is bel< 
it. (Dacca. B.Sc., 1930.) 






EXAMPLES 

30. The latent heat of steam at 100°C. is 536, calculate what 
fraction of this heat is used up in performing external work during 
vaporisation, assuming the density of steam at 100°C to be 0,007 
and an atmosphere to be 10 c dynes per sq. cm. (Man eh-, B.Sc.) 

31. Define the 'triple point' and show that the steam line, hoar- 
frost line and the ice-line must meet in a single point. Draw the 
isodiennals of water for temperatures lower than that of the triple 
point. (A. U., B.Sc, 1934.) 

$2. Define Entropy. What is its physical significance? Show 
that the entropy remains constant in a reversible process but In- 
creases in an irreversible one. (A. U., B.Sc., 1936.) 

S3. Define Entropy. Prove from the principles of thermody- 
namics that the decrease of entropy per unit increase of pressure 
during an isothermal transformation is equal to the increase of volume 
per unit increase of temperature under constant pressure. 

Hence show that heat is generated when a substance, which 
expands on heating, is compressed (A. U., B<5c, 1938.) 

34. What do you understand by 'entropy' ? State the second 
law of thermodynamics in terms of entropy. 

Calculate the change in entropy when m grams of a liquid of 
specific heat s are heated from 2V to r 2 °, and" then converted into 
its vapour without raising its temperature. [Latent heat at 7V =L.] 
(A. IP, B.Sc,, 1941.) 

35. Prove the latent equation 

dL L 

dT T ~ H H 

Calculate the specific heat of saturated steam from the above equa- 
tion and explain the meaning of its negative: value. Given L = 539.3 

cal., T = 100°C ; d h, = - 0.640, c t — 1.01. (Allahabad Univ., 1955 ; 
at 

Baroda Univ., 1953 ; Nagpur Univ., 1956.) 

XT 

1. State Newton's law of cooling. A copper calorimeter weigh- 
ing 15 gms. is filled first with water and then with a liquid. The 
times taken in the two cases to cool from 65°C to 60°C. are 1 70 
sec. and 150 sec. respectively. The weight of the water is 11 gms. 
and that of the liquid is 13 gms. Calculate the specific heat, of the 
liquid. The specific heat of copper ia 0.1. (Cal., B.Sc.) 

2. State Stefan's law and discuss in the light of the same (I) 
Newton's law of cooling, (2) the temperature of the sun, and (3) the 

perature of a tungsten arc. (A. U., B.Sc, 1932.) 

3. State and deduce KirchhofFs- law on the emission and absorp- 
tion of thermal radiation. (Sheff,, B.Sc.) 

4. What is a black body ? What are the characteristics of a 









348 



EXAMPLES 



blackbody radiation? How has it been realised In practice? Des- 
cribe how Stefan's law has been verified w T ich it. (Allahabad Univ., 
1949; Punjab Univ., 1961.) 

5. How can you show that the radiation from an enclosure 
depends only on the temperature and not on the materials of its 
walls. 

Describe a radiation pyrometer, (Delhi Univ., 1955.) 

6. Explain the terms 'emissive power" and 'absorptive power'. 
Deduce that at any temperature the ratio of the 'emissive power to the 
absorptive power 'of a substance is constant and is equal to the 
emissive power of a perfectly black body. (Baroda Univ., 1951.) 

7. Discuss the evidence both theoretical and experimental show* 
ing that good emitters are good absorbers. (Punjab Univ., 1953.) 

8. Explain what you understand by a black body. State 
Stefan's law of radiation and prove, it from thermodynamkal consi- 
deration. Indicate how it can be verified. (Punjab Univ., 1944, 
1950; Roorkee Univ., 1951, 1959.) 

9. Explain what is meant by a "perfectly black body". Write 
a short account of the distribution of energy in the spectrum of the 
radiation from such a body. (Leeds, BJSe.) 

10. Describe any furnace and explain how you would measure 
its temperature, (A. U„ B.Sc) 

11. Discuss the principles underlying the measurement of tern- 

inrii-nT pyrometers, and show how the value 
of the temperature is estimated. Describe fully and clearly a prac- 
tical form of apparatus of this type. (A. U., Br.Sc, lions., 1930.) 

12. DeOne solar constant. Explain with necessary theory how 
the solar constant, is determined. How is the temperature of the 

estimated from the data of the solar constant ? (Allahabad Univ., 
i Punjab Univ., 1952; Nagpur Univ., 1953.) 

13. Under what conditions may a thermodynamic investigation 
of the radiation within an enclosure be carried out? 

Show that under these conditions the radiation varies as the 
fourth power of the temperature, (A. U„ B.Sc*, Hons,, 1928.) 

14. Describe the construction of Boys' radio-micrometer and 
explain the principles involved in its action. (London, B,Sc) 

15. Calculate the temperature of the Earth, assuming that it 
absorbs half the energy falling on it from the sun and that the sun 
radiates as a black body, (Radius of Sun -7x 10 10 cm ; radius of 
Earth — 0,3 >( 10 s cm; mean radius of orbit of Earth = 1,5 X 1& 18 cm J 
temperature of sun's surface— 6Q00°K; SteFan's constant <rz= 5,7 X 10 -8 
C.G.S. units,) 

16. State Stefan-Boltzmann law of radiation and describe 
briefly experiments by which it has been confirmed. (A. U., B,Sc„ 

1934.) 









EXAMPLES 



349 



(in) 

(y) 

(vi) 

(vii) 



17. What is a perfectly black body ? How can such a body be 
realised in practice ? 

How will you verify experimentally that the ratio of the emissive 
and absorptive powers k the same for all bodies and is equal to the 
emissive power of a perfectly black body? (A. U., B.Sc, 1940.) 

18. State Newton's law of cooling mentioning its limitations. 
How would you verify it experimentally ? 

\ body initially at 8D°C cools to 64°C in 5 minutes and to 52°< 
in 20 minutes. What will be its temperature after 16 minutes and 
what is the temperature of its surroundings? (Punjab Univ., 19oI, 
1952 ; Patna Univ., 1951,) 

19. Describe the radiation method of measuring high tempe- 
ratures. (Delhi Univ., 1951.) 

20. Write short notes on the following :— 
(i) Prevost's theory ot exchange. (Punjab Univ., 195 1.) , 

Perfectly black body and its actual realisation. (Delhi 

Univ., 1952,) 
Blackbody radiation. (Delhi Univ., 195G.) 
Stefan's law. (Delhi Univ., 1955,) 
Solar constant. (Delhi Univ., 1956.) 
Determination of the temperature of the sun, (Delhi 

Univ., 1952.) 
Optical pyrometers. (Allahabad Univ.) 

XII 

1. What is meant by (a) relative humidity, (b) dew-point ? 
Describe the Rcgnault's hygrometer and explain how relative 

humidity can be measured with its help. 

2, Explain the general principles underlying the use of a wet 
and dry bulb hygrometer for determining the hygrometric state of 
the atmosphere. 

3. Describe the hair hygrometer and state its uses. 

4, Assuming convective equilibrium of the troposphere, find 
an expression for the decrease in temperature as we go upwards in 
the atmosphere. 

ANSWERS TO NUMERICAL EXAMPLES 



2, 50°C. 



I 
3. 1.55:1. 



4. 1.29 X 10» ei j 



II 



2. 8-22 min. nearly, 

G. 4.217 X ™ 7 ergs/cal 

9. $.89 cal. per gm. per °C. 

10. (a) ; (b) 412.8°C, 

11. -)5.9°C. 



13. 51,8°C 

17. 228°C, 

20. 1.60. 

21. 1.402 






350 






EXAMPLES 
III 



1. 4.22 >< 10 T ergs. 

3. 1.91 X 10° cais. 

5. 4,16 x 10 T ergs. 

7. 40.5 cat, 

II. 3.47 X 10* cro./sec, 

15. 5.67X I" 10 ergs. 



2, 29,5 cal. 

i 9.35 h.p. ; 13.1 °C. per min. 

6. 4.116 X 10 7 ergs. 

10. 0.119°C. 

14. 4.66 X 10* cm./sec. 



IV 



3. 33,6 °C. {a arid h are given in atmospheres and c.c. respec- 
tively for 1 ex. of gas at N.T.P.). 



4. 100.5 cal. per gm. 



V 



VII 



2. 475X10-*. 

t. 59.6 sec. 

10. 0.000801 sq. cm. 



3. 5.646 sec. per day gain per °C. rise, 
6. 0.8 X 10- 4 per °C. 7. ■ JL 

II. 1.20:1 

VIII 

3. 11.3 X 1 8 cal. 

0.39 cm. per hour; 19 hours 
40 minutes. 
14. 1.31 X 10- 3 * 15. 1.26 X 10 B C.G.S. units, 

3.54 X 10~*. 1 !i, I . ] 3 x 10-* C.G.S. units, 

20. 64.8°C 



X 



i. 



6.28 X 10 s ergs. 12, 1.385JJ. 13. 0.36 unite, 
21. O.OOSPC, 23. 0.0252°C. 24. 1674 c.c 

25. 80,8 calories. 26, 0.36°C, 30. 35. -1.07. 



1 
19. 0.36* 



XI 



1. 0.733. 



15. 290°K, 



18. 43°C : 16°a 







The Gas Constant 



Coefficient of expansion for 

perfect gas at 0°C. 
Ice point 
Volume of one gram-molecule 

at N, T. P. = 
Avogadro number N = 

Loschmidt number n = 

Mass of H 1 atom M = 

Standard gravity g s= 

Density of mercury at N.T.P. = 
Standard atmosphere s= 

Mechanical equivalent of I 
Boltzmann's constant k = 

Stefan-Boltzmann constant a— 

Wien's constant b == 

Planck's constant k := 

Velocity of light in vacuum 

Electronic charge <■ ==r 



Faraday number 

Mass of the electron m 

ific electronic charge e/m 

Gravitation constant 



PHYSICAL CONSTANTS* 

R = (8.31436 ±0.00038) X 10 T ergs. 
deg- 1 mole- 1 
= 1.98646 =h 0.00021 cal. deg- 1 
mole— 1 . 

0.0036608 per °C. 
■273.16 ihO.OPK. 

; 22.414 litres 

: (6.0228 ± 0,0011) X 10 28 mole- 1 
(2.6870 =fc 0.0005) X 10 ,ft cm,-'' 
(I.67S39 ± 0.00031) X 10" 2 ' 1 gm* 
980.665 cm. sec.- 2 
13,59504 ± 0.00005 gm. cm-* 
1,013246 X 10* dynes cm.- 2 
,i 04) X ™ 7 ergs cal.- 1 

(1.38047 ± 0.00026) X 10-" ergs. 

(5.672 ± 0.003) X 10" 6 erg. cm- 2 

deg.-" 1 sec. -1 
0.28971 ± 0.00007 cm. degree 
(6.624 ±. 0.002) x 10 ~ 2V erg, sec. 
(2,99776 ± 0.00004) X 10 10 cm. 

c — (2.99776 ± 0.0010) X *0- 1( > e.s.u. 
= (1.60203 =b 0.00034) X 10~ 20 
e.m.u. 
V = 96501 dr 10 int. coul. per gm. 
equiv. 
(9.1066 ± 0.0032) X 1() ~ 2B gm. 

) xlO 7 e, m. n. 
per gm. 
G = (6.670 ± 0.005) X lO -8 dyne cm. 2 
em. -3 



* The values given here are taken from Birue, Reviews of Modern 

s, Vol. 13, p. 233 (1941) ; Amer. Jour! Phys., Vol. IS, p. 63 

5). All quantities in this table involving the mole or the gram 

equivalent are on the chemical scale of atomic weight (O = 16.0000) . 



351 












/ 



SUBJECT INDEX. 

The numbers refer to Pages 



Absolute expansion of liquids, 168 

ilute null point, see Absolute zero, 
lute pyrheliomcter, 312 

Absolute scale of tempera ture, 249 

Absolute zero, 5, 2S0 

Absorption coefficient, 286 

Absorption freezing machines, 127, 131 

Absorption of radiation, 28fi, 289 

Absorptive power, 286, 289 

:.Ljc change of humid air, 325 

Adiabatic demagnetisation, 147 

— equilibrium, 318 
Adiabatic expansion method 

ing % 53 
Adiabatic expansion of compressed 

gases, 132 
Adiabatic stretching of wires, 265 
Adiabatic transformations, 4R 
Adsorption, 125, 133 
Air ci :- 151 et . 

— machine, 154 

151 

liquefacti ictton 

air. 

mtial, 2?8 

1 1:j 

its, 97 
machines, 131 

machines, 

Amor lids, 105 

Andrews' experiments ■ ctivity 

of gases, 198 
Andrews' experiments on carbon 

dioxide, 88 

s experiments, 187 

isotropic bodies, thermal expansion 

of, 165 

application of Klrcbhoff's 

law to, 291 
Athermancy, 288 
Atmospberej thermodynamics of, 316 

pi seq. 
— , distribution of temperature in, 316, 

319 
— , distribution of pressure in, 318 
— t water vapour in, 319 
ispherfc engine, 206 
Atomic energy, 79 



Atomic heat, 43 

— , variation with temperature, 45 
Available energy, 243, 
Ige velocity, 78 
Avogaiiro's law, 75 
— number, 76 



B 



Bartoli's proof of radiation pressure,, 

301 
Baths, fixed temperature, 11 

— . sulphur, 12 

Bcckmann's thermometer, 3 

Becquerel effect, 21 

Bell-Coleman refrigerator, 133 

Berthelot's apparatus, 118 

Bimetallic thermo-rcgulator, 174 

Blackbody, 

— , absorptive power of, 286, 290 

— , definition of, 286 

— , : i —wer of, 290 

— of Fery, 297 

— oi Wien, 296 

— , spectrum, emitted by, 305 
— , total radiation from, 302 
Blackbody curves, 305 
Blackbody radiation, 295, 302 
Blackbody temperature, 308 
■ — , measurement of, 307 et seq. 

— of tlie sun, 314 

point of water, 11 
— , variation with pressure of, 263 
Bolometer, 299 
— , linear 299 
— , surface, 299 
Bo nb calorimeter, 62 
Boyle's law, 5 

— , deduction from kinetic theory, 75 
— >, deviations from, 87 
Bridges for resistance thermometer, 15 
Broadcasting waves, 284 
Brownian movement, 71 
Buusen'iH calorimeter, 33 



Caikndar and Griffiths' bridge, 16 
Callendar's continuous flow calorimeter, 

38 
Caloric theory, 64 






Calorie, definition 

Calorimeter, 

— , bomb, 62 

— , Bunseii's see, 33 

— j continuous flow, 38 

— j copper block, 31 

— , differential steam, 36 

— .i July's steam, 34 

— , Nernst vacuum, 41 

— , steady-flow, ■■'.. 

Calfcrimetry, Chapter II 

— , electrical methods, 37 et seq, 

— , method oi cooling, 32 

— , method of mixtures, 29 

— , methods based on change of state, 

33 
Carbon dioxide, critical constants of, 

90, 102 
— , isothermal curves of, 89 
— , theoretical curves for, 93 
Caruot's cycle, 213 *f seq. 
i hot's engine, 213 et seq. 
. reversibility of, 218 
; theorem, 218 
Cascade process of refrigeration, 135 
Change of state, Chapter V 
— , application of thermodynamics to, 

262, 272 
Charles' law, 5, 176 
Chemical thermometers, 3 
Claude's air liqucfier, 143 
Claude-Heylandt system, 144 
,. Clausius-Oapeyron equation, 262 
— , Clapeyron's deduction of, 272 
Clement and Desormes* apparatus, 53 

leal thermometer, 3 
Coefficient of performance, 235 
Combustion engines, see Internal com- 
bustion engines 
Comfort chart, 151 
Comfort zone, 152 
Comparator method, 159 
Compensation of clocks and watches, 

172 
Compensation of mercury pendulum, 

171 
Compound engine, 212 

idensation, methods of causing, 324 
Conduction of heat, Chapter VIII 
— , in three dimensions, 191 
— r Kinetic Theory of, 84 
— through composite walls, 190 
Conductivity, thermal, definition of, 

179 
— . of different kinds of matter, 178 
23 



, relation between electrical 

tivity and, 190 
Conductivity of Earth's crust, 188 
Conductivity of gases, S4, 197 ei seq. 
— , relation between, viscosity and, 84 
— , variation with pressure, 84, 199 
Conductivity of glass, 194 
Conductivity (thermal) of liquids, 196 

et seq. 
Conductivity (thermal) of metals, 

determination, 180 et seq. 
— , by a combination of steady and 

variable flow, 186 

— from calorimetric measurement, 180 

— from periodic flow of heat, 187 

— from temperature measurements, 182 
Conductivity of poorly conducting- 

solids, 192 et seq. 
Conductivity of rubber, 194 
Conductivity, thermometric, 182 
Constant, Boltzmann, 76 
— , critical, see Critical constants 
— , gas, 6, 10 
— , Planck's, 304, 351 
— , solar, 311 
— t Stefan's, 302 
— , Wien's, 304, 306, 351 
Constant flow method for gases, SI 

— for liquids, 38 

Constants in kinetic theory, 85 
Continuity of spectrum, 281 
Continuity of states, 90 
Convection of heat, 201 
— , forced, 203 
— , natural, 202 
Convective equilibrium, 318 
Cooling by adiabation expansion, 132 

— due to desorption, 133 

— due to Joule-Thomson effect, 136 
et seq. 

— due to Peltier effect, 133 
— , Ncwton'slaw of, 287 

— , regenerative, 141 

Correction for emergent column, 171 

Correction of barometric reading, 170 

Correction of gas thermometer from 
Joule-Thomson effect, 269 

Crank, 210 

Critical coefficients, 96 

Critical constants, 90 

— , deduction from van der Waals' 
■:viriti;,n, •'.!.'< 

— , determination of, from law of recti- 
linear diameters, 100 

— , tabic of, 102 

Critical density, 101 




subject iNnnx 



355 



Critical point, 90 

— ., matter near the, 102 

Critical pi i 95, 100 

— temperature, 90, 95, 100 

— volume, 90, 95, 100 
CryophortiK, 126 
Cryustatg, 149 

Crystals, expansion of, 165 

Cycle, Carfiot, 214 

— — , efficiency of, 216 

—j reversibility of I 

— , Diesel, 226 

— »— , efficiency of, 227 

— Otto, 223 

, efficiency of, 224 

—, Rankine's, 220 

, efficiency of, 221 

Cylindrical shell method o| finding 

CI iiiuCtivLLy, 193 



I J 



Dalton's law of partial pressures, 75 

■ ty lamp, 
Dead space correction, 9 

.7'' 

■,'apour, m Vapour density 
• from I ! ■;■ Ic's law, 

— sei 
Die 

: air ili: • . 278 

Diffuse radiati n, energy density of, 301 
— , pressure of, 301 

meter method, lci7 
Disappearii ient pyrometer, 310 

: ir poor conductors, 194 
tti energy, 245 
Displacement law, Wien's, 304 
Distribution of energy in blackbody 

rum, 
Distril (Maxwell . 

ting engine, 208 
ie's theory of conduction, 191 
D-slide valve, 209 
Duiong and Fctit's law, 43, 81 
— , from kinetic stand-paint, 81 
— , illustration of, -ii 
Dynamic method of finding vapour 
pressure, 114 



Earth, temperature inside the, 188 

Earth's crust, conductivity of, 188 

Eccentric, 21 1 

Effective temperature, 153 

Efficiency of engines, 212 

— , Camot's cycle, 216 

— , Diesel cycle, 

— , Otto cycle, 225 

l\]i.i.:trical methods in Calorimetry, 37 

et seq. 
nprir "magnetic waves 283*234 
Emission coefficient, 290 
— , total, 302 
Emissive power, 290 
Energy, conservation of, 242 
— , chscviiuiHHMK changes in, 245 
— , dissipation of, 343 
— , distribution of, in the spectrum, see 
Distribution of energy in the black- 
body spectrum 
— , forms of, 241 
— , molecular and atomic, 79 
— , transmutation of. 242 
Engine, Carnot's, 213 
— , Diesel, 226 
— , hot-bulb, 228 
— , Internal combustion, see internal 

combustion en 
— , jet, 234 
— , National gas, 228 
— , OtitO, 223 

— i re ; irreversible, 218 

— , semi-Diesel or hot-bull i. 

, single-acting, 209 
Entropy, 25 1 et seq. 

rgy, 256 
— , change of, in reversible pp ■ 253 

— — , in irreversible pr icesse . 254 
i definition of, 251 

— , law of increase of, 255 

— .if a perfect gas, 2~? 

— of u system, 253 

— of steam, 258 

— , physical concept of, 257 
— -, statement of sec i thermo- 

dynamics, ill t< rn - of, 
Entropy-temperature dlagramj 257 
3 T .i|i::iiii:n of Clausius-Clapeyron. 262, 

— of heat conduction (Fourier), 183 

Equati ms of state i p, IV 

— , defmitirn of, 87 

Kperimental study of, 96 

— of van der Waals, 'Jl 



Equilibrium, adiabatic or convective, 318 
— , radiative, 317 
Equipartkion of energy, 79 
Eutcctic mixture, 126 

— temperature. 

Examples on Thermodynamics, 271 
Exchanges, Frevosfa theory of, 286 
Expansion (thermal). Chap. VI f 
— , applications of, 172-174 
Expansion of anisotropic bodies, 165 
Expansion of crystals, 161, 165 
— j, Fizeau's method, 161 
— , fringe-width dilatometer methodj 

163 
Expansion of gases, 175 
— , determination of, volume co- 
efficient of. 175 
Expansion of invar, 165 
i 1 -. | ' •; i.i of liquids, 166 et seq, 
— , absolute, by hydrostatic balance 

method, 169 
— , dilatometer method, 167 

msion of silica, 165 
Expansion (linear) of solids (isotro- 
pic), 157 
— , discussion ::i results, 164 
— , earlier measurements of, 158 
— , measurement of, 159 cl seq. 

, comparator method, 159 

■, J .;•:■-■ titer and Laplace's method, 

159 

— relative, hy Helming r S tube method, 
160 

Expansion of water, 171 

Expansion, surface and volume, lb5 



P 



Film method of determining conducti- 
vity of liquids, 196 

— of gases, 198 

First law of thermodynamics, see Ther- 
modynamics, first law of 
Fixed points, chart of, 25-26 
Fixed temperature baths, 11 
Flow of heat (rectilinear), 182 
— , combination of steady and variable, 

18! i 

— , periodic, 187 
■heel, 210 
ies' method, 186 
unhofer lines, 291 

— spectrum of the sun, 291 
Freedom, degrees of, 79 
Free path, 81 

— , mean, see Mean free path 



Freezing mixture, 126 

Freezing point, see Melting point 

Frigidairc, 130 

Fuel in engines, 229 

Fusion, 104 ei seq. 

— , effect of pressure on, 109 

— — , thermodynamic explanation of, 

262 
— , latent heat of, see Latent heat of 
fusion 

- of alloys, 110 



Gamma rays, 285 

Gas engine-. 332 1 1 seq. 

Gas laws. 5 

— , deduction of, from kinetic theory, 
7B 

— , deviation of gases from, 87 et seq. 
perfect, 9, 87 

Gas scale corrections from Joule- 
Thomson effect, 269 

Gas scale, perfect, 9 

Gas thermometers, 4 

— , Calendar's compensated air thermo- 
meter, 6 

— , constant pressure, 6 

— , constant volume, 6, 7 

— , constant volume hydrogen, 7 

— , standard, 7 

Gases, conductivity of. 1 r 7 

— , equation of state for, Chap, IV 

— , expansion of, Chap. I & VII 

— , liquefaction of, 133 et seq. 

Gases, permanent, 194 

Gases, specific heat of, 49 et seq. See 
also Specific heat of gases 

— , thermal conductivity of, 84, 197 

■ — , viscosity of, 83 

Glacier motion, 110 

j, conductivity of, 194 

Governor, 210 

Gruneisen's law, 165 

Guard ring, 181 



H 



Hair hygrometer, 323 
Hampson's air liquefier, 143 
Heat, a kind of motion, 64, 70 

— and light. 281 

— and work, 65, 243 

— as motion of molecules, 70 
— balance in human 








81 BJECT 1NIHCX 






Heat by friction, 6J 

Heat, caloric theory of, 64 

— , convection of, 201 

— , dynamical equivalent of, 6S-70 

— engines, Chap, IX, see also Hngines 

and Cycles 
— , latent, 104, 107, 117 

— — , variation with temperature, 120 
— , nature of, 64 c! scq. 

— of combustion, 62 

— , periodic flow of, 187 at teq, 

— , propagation of, J 78 

— , radiant, Chap, XI, see also 

Radiation. 
— , rectilinear flow of, 182 
— , specific, see Specific heat 
— , steady flow of, 183 
— t unit of, 28 
Helium, liquefaction of, 146 
— , solidification of, 147 
Henning's tube method, 100 
Hertzian waves, 282 
Hess's law, 246 
Hot-bulb engine, 228 
JlnL-wire method of finding conducti- 
vity, 197, m 

Human body, heat balance in, 63 
Humid air, adiabatic change of 
Humidity, 320 
— , absolute, 320 
— , relative, 320 
Hydrogen, conductivity of, 198 
— , critical temperature 
— , Joule-Thomson cooling in, 144, 268 
faction of, 144 

— spectrum, 292 

i crmometer, 7 
Hydrostatic balancv 169 

Hygrometer, 321 
— , chemical, 321 
— , dew-point, 322 
— , hair, 323 

— , Regnauit's dew-point, 
— , wet and dry bulb, 3 J 3 
Hygromctry, 32] 
Hypsometer, 12 



Ice, latent heat of, 105, 107 
Indicator diagram, 213 
Infra-red ray?, 282 
Ingen-Hausz's experiment, 184 
Integrating factor, 240, 260 
Internal combustion engines, 222 el seq. 
-, application of, 229 



— , fuel used in, 229 

• i i * a gas h 
panding, 47 

i Ltional temperature cale, 24 

Intrinsic energy, vnrin: , with 

volume, 266 
Invar, expansion of, 165 
Inversion temperature, I 
— , expression for, from thermo- 
dynamics, 269 
Irreversible engines, 218 
— process, 216 
Isothermais, 89, 98 



J 



J, ili.:'.i.:niiin;-ilii..i:-. of, 66 et scq. 

Jena glass, 2 

Jet propulsion, 234 

Joly's steam calorimeter, 34 

— , differential form, 36 

Joule's experiments. 47, 65 

— law, 47 

— method of finding y, 54 
Joule-Thomson effect, 336. 268 

— , Correction of gas thermometers 
from, 269 

— for gas obeying van der Waals' 

— inversion of, 145 



K 



Kapitza's tiquefier, 144, 147 
Kinetic theory of matter, Chap, II I 
— , constants, table of, 85 
— , deduction of gas laws from, 75 
— , evidence of the molecular agitation, 

71 
— , growth of, 70 

— , introduction of temperature into, 76 
— of specific heat, 79-81 
— , pressure of perfect gas from, 72 
Kirchhofl's explanation of FraunhofLr 

lines, 292 
Kirchhoff's law, 290 
— , application to Astrophysics, 291 

. deduction of, 2<>5 



Langlcy's bolometer, 299 

Laplace's formula, 3 If! 

Lapse-rale, 316 

Latent heat of fusion, 104 

— , determination of, 107 et seq. 



247 

' m Lit. d of finding conductivity of 
ii !■;, 1% 
• ii poor conductors, 
• I h , 278 

ion by heating, 281 
l ,in 1 1 ■'■■ ii achine, 142 
i [near ton, 157 et scq. 

ion of air, 142 
Liquefaction of gases, 133, el 

— by application of Joule -Thomson 

effect, 136 et seq 
- by application of pressure and low 
temperature, 133 

— by cascades or scries refrigeration, 

135 
Liquefaction of helium by IC O lines, 

146 
Liquefaction of hydrogen, 144 
Liquid air, uses of, 149 
Liquids, expansion of, see Expansion 

of liquids 
— , thermal conductivity of, tit Con- 

ivity of li 
Liquid thermometers, 1-4, 23 

of planetary atmosphere, 78 
Low temperature siphons, 149 
Low temperature techniques, 148 
Low temperature thermometry, 22 



M 



Matter, continuity of liquid mid gaseous 

states, 90 
— , state of, near the critical point, 102 
Hatter, three states of, 104 
Maximum and minimum thermometer, 

3 
Maxwell's demon, 256 
Maxwell's law, graphical rcpreseuta- 
on of, 77 

— of distribution of velocity, 76 
Maxwell's tbermodynamical relation- 

ships, 260 et seq. 
Mayer's hypothesis, 47 
Mean free path, 81 

— of Maxwell, 82 _ 
Mean square velocity, 78 
Mean velocity, 78 

Median ir tifvalent of heat, 6fi et 

seq. 
— , value of, 70 
Melting, d electrical resistance 

on, 106 



— , change of vapour pressure on, 106 
— , chat • volume on, 106 
Melting point, 106 

— of ice, effect of pressure on, 109, 
262 

— of metals, 106 

, effect of pressure on, 263 

Mercury, conductivity of, 181 
— , expansion of, 170 
Mercury thermometer 
— , errors oi, 

, correction for, 

Method, electrical, of measuring specific 
heat, 37 et seq, 

— of finding latent heat, 108 

— of cooling, 32 

— of melting 

— of mixtures, 29, 49 
ant! latent heat of fusi' 

107 

— — and latent heat of fusion of 

metals, 108 
Methods of causing condensation. 
Molecular I .mi, 44, 79 
Molecules, diameter of, 81 
— , free path of, see Mean free path 
— , translation;)! energy of, in gas, 76 
— > velocity of, in pas, see Velod 
Multiple expansion engine, 212 



K 



National gas engine, 228 

Natur> :, 64 

Ner nst's copper block calorimeter, 31 

Mernst's vacuum calorimeter, 41 

Neumann's law of molecular heats, 44 

Newton's law of cooling, 29, 287 

Nozzle, expansion through, 230 







Optical method of measuring expan- 
sion, 161-164 

Optical pyromctry, 309 see also Radia- 
tion pyrometry 

Otto cycle, 223 

Otto engine, 223, 229 



Path, see Mean free path 
Peltier effect, 21 
— , cooling due to. 133 
Pendulum, gridiron, 173 
— , mercury, 171 



$58 



SUBJECT INDEX 












Perfect differential, 239 

Perfect gas, 5, 87 
— t pressure of, 72 

— scale, 9 

Periodic flow of li.-;it, 187 

Permanent gases, 134 

— , liquefaction of, see Liquefaction nf 

Perpetual motion 

— of first kind, 219, 

— of second kind, 219, 248 
Fetrolei m ether thermometer, 23 
Phenomena, mean free path, 81 
— , transport, S2 

Phenomenon of conduction. 178 (Chan. 
VIII) 

— of viscosity, S3 
Photosphere, 314 

— , temperature of, 314 

Planck's radiation formula, 304 

Planet, artificial, 23S 

Planetary atmosphere, loss of, 78 

Platinum resistance thermometer, 13-18 

Poro. :periment, 139 

Potentiometer, 20 

p. 97 
Pre::. ' 7n 

30© 

see 

ure 

Priji 

Principle of regenerative 141 

iPyrhel I lute, 312 

— , v, 

— , wi tir, 312 

Pyrometry, 306 si 

— » fPs pyrometers, 306 

— , optical, 309, see also Radiation 

pyp i i 
— , radiation, 307 
— , resistance, 307 
— , thermo-electric, 307 



Quantity of heat, 28 
Quartz, 165 

R 

iant energy, identity of light and, 
281 e! seq. 



— , nature of, 279 

— , passage through matter of, 28fi, 288 

— , properties of, 279 

Radiatioi . I . a XI 

— , application of thermodynamics to, 

see Therm- dynamics of radiation 
— , blackbody, 293-297 
, analogy between perfect gas and, 

301 
Radiation constant, Planck's, 304 
— , Stefan's, 302 
— , Wien's. 304 
Radiation correction, 29 
Radiation, diffuse, 301 

— from the stars, 314 
Radiation laws, Planck's law, 304 
— , Stefan-Boltzmanft's law, ii 
Radiation, measurement of, 297 et seq. 
Radiation, passage through matter of. 

Radiation, pressure of, 300 

— , properties and nature of, 279 

— , temperature, 293 

Radiation pyrometers, 307 et seq, 

liative equilibrium, 317 
Radiometer, Crookcs', 298 
i meters, 297 ei seq, 
Radiomicrometcr, 300 
220 
', 221 
i of the sped 

.■rq. 

—, adiabatic expansion method (Cle- 
ment and Desormes) , 53 
— , experiments of Partington, SS 
— , method of Ruchardt, 56 
— , table of results, 62 
— , velocity of sound method, 57 
Reaction turbines, 233 
Rectilinear diameters, law of, 100 
Rectilinear flow of heat, 182 
Reflecting power, 288 

128 
— , characteristics of, 129 
Refrigerating machine, absorption, 131 
— s air compression, 133 

— Bell-Coleman, 133 
— , Carnol, 218 

, efficiency of, 235 

— , vapour compression, 127 
——j efficiency of, 235 
Refrigeration, Chap. VI 
— , cascade process of, 135 

— due to Pettier effect, 133 
— , principles used in, 125 



by adding a sal , 125 

b i • der n duced 

i VH |i|i:i||, 

- by Joule Thomson expo 

Refrigci mi Refrigerating machine 

i : u i . i '.••: OOOltng, 141 
. thermal, 179 
:e thermometry, 13, 23, ■'<'•'■' 
, platinum, 13 

engines, 218 
— process, 216 

cket, 234 
Room 154 

i mean square velocity, 74 
Rowland's experiments, 66 
n I icperiments, 56 



..Ivc, 206 

Satellites artificial, 2 

Saturated vapour, 111 

— , density of, 123 

— , specific heat of, 273 

— , vapour pressure of, 112-117 

Searle's apparatus for conductivity, 
ISO 

Secondary thermometers, 10, 27 
. 228 

Single-acting engine, 209 

Slide valve, 209 

Solar constant, definition of, 311 

— , determination of, by absolute pyr- 
imeter, 312 

Solidification of helium, 147 

Solids, conductivity of, see Conducti- 
vity of solids 

Sound, velocity of, 57 et seq. 

Specific beat, definition of, 

— , difference between the two, 46, 245, 

— , l-:!n ■ '' 

- ~, methods of measurement. Chap, II 

— , negative, 274 

Specific heats of gases, 45 et seq. 

Specific: heat of gases, determ'nati n 

(experimental) at constant pressure, 

49 et seq. 
— , at constant volume, 51 et seq. 



■ ! -cory 

Specifn heat of liquids, 31, 37-40 

d, 37 
— , method of mixtures, 29 
— , stead j- -flow electric calorimeter, 38 
heat of vapour, saturated, 273 

— unsaturated, 61 

Specific heat of solids, 30, 41 et seq. 
— , electrical method, 41-43 
— v method of mixtures, 29 
— i variation with temperature. 
Specific heat of superheated vapour, 61 

• , 40 
Spectra of stars, 292 

. V 

— -, therm idynamic treatment of, 262, 

272 
States of .104 

Steam calorimeter (Joly), 34 et seq. 
, 205 ei sr ; 

— , < 208 

— i, modern, 211 

— , New men's atnn spl &rj 

arts of, 208 ei . 
— , Savi py's, 206 
— , single-acting, 209 
Steam, enti 
— , expansive use of, 209 

— total beat of 

ii jet, theory of, 230 

, 229 
mp ,23; 
uf Curtis, 232 

— of Dc Laval, 231 

— of Parsons, 233 
— , reaction, 2 
Stefati-iBoltzmaiui'a lav,-, 302 

— , experimental verification of, 302 
Stefan's constant, : : 
Stral here, 317 
— , pressure did : n in, 318 

Stuffing box, 209 
Sublimation, 105 
Sulphur boiling apparatus, 12 
Sun, radiation from, 311 et seq. 
— , temperature of i 

iling, 111 
h a ting, 111 

, 61 
259 



360 



SUJJJIiCT INDEX 



Temperature, absolute, 10, 249 

— baths, 11 

-, critical, 90, 95, 100 

— — , table of, 136 

Temperature, critical, value_ of, from 

ve.jj tlur Waals' equation, 95 
— , effective, 153 
— , definition of, 1 
— , distribution of, in the atmosphere, 

316 
_ gradient, 319 
— , high, measurement of, 306 
— , low, measurement of, 22 
„ — t production of, Chap. VI 

— of inversion, 145, 268 

— of stars, 314 

— of sun, 314 

— radiation, 293 

— scale, international, 24 
— . standard, 11 

— , underground, 188 

— wave, wavelength of, 188 
Theorem, Carnot's, 218 
Theory, atomic, 70 

— , kinetic, of matter, Chap. Ill 
— , molecular, 70 

— of exchanges (Frevost's), 286 

Thermal definition of, 23 

Xhet wt Condu.- 

(the i"' . , 

— , ratio of electrical conductivity to., 

190 
Thermal exp slon 

banco, 179 
Thermal state of a bod: i 

rmo-enuples, 18 et se<"!. _ 
Thermodynamic relationships (3 

well), 260 ei 
— , first relation, 2 

, application to change of freez- 
ing' point by pressure, 262 

} application to liquid film, 264 

_ second relation, 264 

— , third and fourth relations, 266 

Thermodynamic scale of temperature, 

10, 249 
Thermodynamica! variable, 237 
Thermodynamics, application ot, to 

change of state, 262, 272 
Thermodynamics, application of, to 

radiation, see Thermodynamics of 

radiation 
Thermodynamics, examples on, 271^ 
Thermodynamics, first Jaw of, 243, 259 



— of evaporation, 263 

— of fusion, 262 
Thermodynamics of radiation, 293 et 

seq. 
Thermodynamics of refrigeration, 235 
Thermodynamics of the atmosphere, 316 
Thermodynamics, scope of, 237 
— , second law of, 247 
Thermodynamics, second law of, 
— ? Clausius J s enunciation of, 248 

, Kelvin's enunciation of, 248 

— , preliminary statement of, 243 
— , scope of, 247 

— statement of, in terms of entropy, 
255 

Thermo-electric thermometry, 18 et seq. 

Thermometer, alcohol, 2, 23 

— , Beckmatm, 3 

— , Callendar's compensated air, 6 

— , chemical, 3 

— , clinical, 3 ^ 

— , constant volume hydrogen, 7 

— , gas, 4 et seq. 

— , liquid, 1-3 

— , maximum and minimum, 3 

— , mercury', 1 

—j petroleum ether, 23 

— , platinum, 13 et seq, 

—, secondary, 10, 27 

— , standard gas, 7 

— , standardisation of, 10 

— , thermo-electric, 18 et seq. 

— , vapour pressure, 22, 23 

_, weight, 167 

Thertnometric conductivity, 182 

Thcrrnometr] , Char*. I 

— , high temperature, 306 el seq. 

— , low temperature, 22 

— , resistance, 13 et seq. 

— , thermo-electric, 18 *-'-' seq. 

Thermopile, 279 

Thcrmo-regulator, 174 

Thermostat, 174 

Throttle valve, 210 

Toluene thermostat, 174 

Total heat of steam, 221 

Total radiation pyrometers, 308 

Transformation, adiabatic, 4% 

Transport phenomena, 82 

Triple point, 275 

— for water, 275 
Tropopausc, 317 

— height of, 317 
Troposphere, 317 

— , pressure distribution in, 318 

— ■ temperature distribution in, 319 









t'urliin 

U 

i :gy, 256 

i: round temperature, 188 
tjaii of heat, 



linn, production of high, 149 
• lr:r Walls' equation of state, 91 
. critical constants from, 95 

— , deduction of, 91 

-— , defects in, 96 

— , discussion of, 9S 

— , Joule-Thomson effect from, 140, 
268 

— , methods of finding a and b, 92 

Vaporisation, 104, 111 et seq, 

— , latent heat of. 117 see also Latent 
heat of vaporisation 

Vapour compression machine, 127 

— , efficiency of, 235 

Vapour density, 122 et seq. 

— of saturated vapour, 123 

— , Victor Meyer's method, 123 

Vapour pressure curve for water, 102, 
321 

— , discussion of results, 116 

— , laws of, for mixture of liquids, 116 

Vapour pressure measurements, dyna- 
mic or boiling point method, 114 

— , static method, 113 

Vapour pressure of water, 112 

— over carved surfaces, 117 

Vapour pressure thermometers, 22, 24 

Velocity, average, 78 

— , law of distribution of, 76 



t probable 

square, 74 

ity of sound, 57 et seq. 
Vibrational motion of diatomic mole- 
cules, 80 
Viscosity, 83 

— , discussion of the result, 84 



W 



Water, boiling point of, 11 

— equivalent, 28 

— , expansion of, 171 

— , freezing point of, on absolute scale, 

10 
— , specific heat of, 40 

— vapour in the atmosphere, 319 

, method uf condensing, 324 

— , vapour pressure of, 320 
Watt's double-acting engine, 208 

— experimental condenser, 208 

— governor, 210 

— inventions, 207 et seq. 
Weight thermometer, 167 
Wiedemann and Franz's law, 190 
W ion's constant, 304 

Wien r s displacement law, 304 

— , experimental verification of, 306 

Work, graphical representation of, 213 

— obtained in isothermal and adiabatic 
expansion, 245 

Working substance, 214 



X-rays, 285 



Zero, absolute, 10, 250